diff --git a/t/t5558-clone-bundle-uri.sh b/t/t5558-clone-bundle-uri.sh
index 3816ed5058d..33a7009e9a2 100755
--- a/t/t5558-clone-bundle-uri.sh
+++ b/t/t5558-clone-bundle-uri.sh
@@ -58,7 +58,7 @@ test_expect_success 'create bundle' '
 test_expect_success 'clone with path bundle' '
 	git clone --bundle-uri="clone-from/B.bundle" \
 		clone-from clone-path &&
-	git -C clone-path rev-parse refs/bundles/topic >actual &&
+	git -C clone-path rev-parse refs/bundles/heads/topic >actual &&
 	git -C clone-from rev-parse topic >expect &&
 	test_cmp expect actual
 '
@@ -68,9 +68,9 @@ test_expect_success 'clone with bundle that has bad header' '
 	git clone --bundle-uri="clone-from/bad-header.bundle" \
 		clone-from clone-bad-header 2>err &&
 	commit_b=$(git -C clone-from rev-parse B) &&
-	test_grep "trying to write ref '\''refs/bundles/topic'\'' with nonexistent object $commit_b" err &&
+	test_grep "trying to write ref '\''refs/bundles/heads/topic'\'' with nonexistent object $commit_b" err &&
 	git -C clone-bad-header for-each-ref --format="%(refname)" >refs &&
-	test_grep ! "refs/bundles/" refs
+	test_grep ! "refs/bundles/heads/" refs
 '
 
 test_expect_success 'clone with bundle that has bad object' '
@@ -78,8 +78,8 @@ test_expect_success 'clone with bundle that has bad object' '
 	git clone --bundle-uri="clone-from/bad-object.bundle" \
 		clone-from clone-bad-object-no-fsck &&
 	git -C clone-bad-object-no-fsck for-each-ref --format="%(refname)" >refs &&
-	grep "refs/bundles/" refs >actual &&
-	test_write_lines refs/bundles/bad >expect &&
+	grep "refs/bundles/heads/" refs >actual &&
+	test_write_lines refs/bundles/heads/bad >expect &&
 	test_cmp expect actual &&
 
 	# Unbundle fails with fsckObjects set true, but clone can still proceed.
@@ -87,14 +87,14 @@ test_expect_success 'clone with bundle that has bad object' '
 		clone-from clone-bad-object-fsck 2>err &&
 	test_grep "missingEmail" err &&
 	git -C clone-bad-object-fsck for-each-ref --format="%(refname)" >refs &&
-	test_grep ! "refs/bundles/" refs
+	test_grep ! "refs/bundles/heads/" refs
 '
 
 test_expect_success 'clone with path bundle and non-default hash' '
 	test_when_finished "rm -rf clone-path-non-default-hash" &&
 	GIT_DEFAULT_HASH=sha256 git clone --bundle-uri="clone-from/B.bundle" \
 		clone-from clone-path-non-default-hash &&
-	git -C clone-path-non-default-hash rev-parse refs/bundles/topic >actual &&
+	git -C clone-path-non-default-hash rev-parse refs/bundles/heads/topic >actual &&
 	git -C clone-from rev-parse topic >expect &&
 	test_cmp expect actual
 '
@@ -102,7 +102,7 @@ test_expect_success 'clone with path bundle and non-default hash' '
 test_expect_success 'clone with file:// bundle' '
 	git clone --bundle-uri="file://$(pwd)/clone-from/B.bundle" \
 		clone-from clone-file &&
-	git -C clone-file rev-parse refs/bundles/topic >actual &&
+	git -C clone-file rev-parse refs/bundles/heads/topic >actual &&
 	git -C clone-from rev-parse topic >expect &&
 	test_cmp expect actual
 '
@@ -173,12 +173,12 @@ test_expect_success 'clone bundle list (file, no heuristic)' '
 	git -C clone-list-file cat-file --batch-check <oids &&
 
 	git -C clone-list-file for-each-ref --format="%(refname)" >refs &&
-	grep "refs/bundles/" refs >actual &&
+	grep "refs/bundles/heads/" refs >actual &&
 	cat >expect <<-\EOF &&
-	refs/bundles/base
-	refs/bundles/left
-	refs/bundles/merge
-	refs/bundles/right
+	refs/bundles/heads/base
+	refs/bundles/heads/left
+	refs/bundles/heads/merge
+	refs/bundles/heads/right
 	EOF
 	test_cmp expect actual
 '
@@ -220,10 +220,10 @@ test_expect_success 'clone bundle list (file, all mode, some failures)' '
 	git -C clone-all-some cat-file --batch-check <oids &&
 
 	git -C clone-all-some for-each-ref --format="%(refname)" >refs &&
-	grep "refs/bundles/" refs >actual &&
+	grep "refs/bundles/heads/" refs >actual &&
 	cat >expect <<-\EOF &&
-	refs/bundles/base
-	refs/bundles/left
+	refs/bundles/heads/base
+	refs/bundles/heads/left
 	EOF
 	test_cmp expect actual
 '
@@ -253,7 +253,7 @@ test_expect_success 'clone bundle list (file, all mode, all failures)' '
 	git -C clone-all-fail cat-file --batch-check <oids &&
 
 	git -C clone-all-fail for-each-ref --format="%(refname)" >refs &&
-	! grep "refs/bundles/" refs
+	! grep "refs/bundles/heads/" refs
 '
 
 test_expect_success 'clone bundle list (file, any mode)' '
@@ -282,9 +282,9 @@ test_expect_success 'clone bundle list (file, any mode)' '
 	git -C clone-any-file cat-file --batch-check <oids &&
 
 	git -C clone-any-file for-each-ref --format="%(refname)" >refs &&
-	grep "refs/bundles/" refs >actual &&
+	grep "refs/bundles/heads/" refs >actual &&
 	cat >expect <<-\EOF &&
-	refs/bundles/base
+	refs/bundles/heads/base
 	EOF
 	test_cmp expect actual
 '
@@ -313,7 +313,7 @@ test_expect_success 'clone bundle list (file, any mode, all failures)' '
 	git -C clone-any-fail cat-file --batch-check <oids &&
 
 	git -C clone-any-fail for-each-ref --format="%(refname)" >refs &&
-	! grep "refs/bundles/" refs
+	! grep "refs/bundles/heads/" refs
 '
 
 test_expect_success 'negotiation: bundle with part of wanted commits' '
@@ -322,10 +322,10 @@ test_expect_success 'negotiation: bundle with part of wanted commits' '
 	git clone --no-local --bundle-uri="clone-from/A.bundle" \
 		clone-from nego-bundle-part &&
 	git -C nego-bundle-part for-each-ref --format="%(refname)" >refs &&
-	grep "refs/bundles/" refs >actual &&
-	test_write_lines refs/bundles/topic >expect &&
+	grep "refs/bundles/heads/" refs >actual &&
+	test_write_lines refs/bundles/heads/topic >expect &&
 	test_cmp expect actual &&
-	# Ensure that refs/bundles/topic are sent as "have".
+	# Ensure that refs/bundles/heads/topic are sent as "have".
 	tip=$(git -C clone-from rev-parse A) &&
 	test_grep "clone> have $tip" trace-packet.txt
 '
@@ -337,8 +337,8 @@ test_expect_success 'negotiation: bundle with all wanted commits' '
 		--bundle-uri="clone-from/B.bundle" \
 		clone-from nego-bundle-all &&
 	git -C nego-bundle-all for-each-ref --format="%(refname)" >refs &&
-	grep "refs/bundles/" refs >actual &&
-	test_write_lines refs/bundles/topic >expect &&
+	grep "refs/bundles/heads/" refs >actual &&
+	test_write_lines refs/bundles/heads/topic >expect &&
 	test_cmp expect actual &&
 	# We already have all needed commits so no "want" needed.
 	test_grep ! "clone> want " trace-packet.txt
@@ -363,13 +363,13 @@ test_expect_success 'negotiation: bundle list (no heuristic)' '
 		clone-from nego-bundle-list-no-heuristic &&
 
 	git -C nego-bundle-list-no-heuristic for-each-ref --format="%(refname)" >refs &&
-	grep "refs/bundles/" refs >actual &&
+	grep "refs/bundles/heads/" refs >actual &&
 	cat >expect <<-\EOF &&
-	refs/bundles/base
-	refs/bundles/left
+	refs/bundles/heads/base
+	refs/bundles/heads/left
 	EOF
 	test_cmp expect actual &&
-	tip=$(git -C nego-bundle-list-no-heuristic rev-parse refs/bundles/left) &&
+	tip=$(git -C nego-bundle-list-no-heuristic rev-parse refs/bundles/heads/left) &&
 	test_grep "clone> have $tip" trace-packet.txt
 '
 
@@ -395,13 +395,13 @@ test_expect_success 'negotiation: bundle list (creationToken)' '
 		clone-from nego-bundle-list-heuristic &&
 
 	git -C nego-bundle-list-heuristic for-each-ref --format="%(refname)" >refs &&
-	grep "refs/bundles/" refs >actual &&
+	grep "refs/bundles/heads/" refs >actual &&
 	cat >expect <<-\EOF &&
-	refs/bundles/base
-	refs/bundles/left
+	refs/bundles/heads/base
+	refs/bundles/heads/left
 	EOF
 	test_cmp expect actual &&
-	tip=$(git -C nego-bundle-list-heuristic rev-parse refs/bundles/left) &&
+	tip=$(git -C nego-bundle-list-heuristic rev-parse refs/bundles/heads/left) &&
 	test_grep "clone> have $tip" trace-packet.txt
 '
 
@@ -428,10 +428,10 @@ test_expect_success 'negotiation: bundle list with all wanted commits' '
 		clone-from nego-bundle-list-all &&
 
 	git -C nego-bundle-list-all for-each-ref --format="%(refname)" >refs &&
-	grep "refs/bundles/" refs >actual &&
+	grep "refs/bundles/heads/" refs >actual &&
 	cat >expect <<-\EOF &&
-	refs/bundles/base
-	refs/bundles/left
+	refs/bundles/heads/base
+	refs/bundles/heads/left
 	EOF
 	test_cmp expect actual &&
 	# We already have all needed commits so no "want" needed.
@@ -465,7 +465,7 @@ test_expect_success 'clone HTTP bundle' '
 
 	git clone --bundle-uri="$HTTPD_URL/B.bundle" \
 		"$HTTPD_URL/smart/fetch.git" clone-http &&
-	git -C clone-http rev-parse refs/bundles/topic >actual &&
+	git -C clone-http rev-parse refs/bundles/heads/topic >actual &&
 	git -C clone-from rev-parse topic >expect &&
 	test_cmp expect actual &&
 
@@ -476,7 +476,7 @@ test_expect_success 'clone HTTP bundle with non-default hash' '
 	test_when_finished "rm -rf clone-http-non-default-hash" &&
 	GIT_DEFAULT_HASH=sha256 git clone --bundle-uri="$HTTPD_URL/B.bundle" \
 		"$HTTPD_URL/smart/fetch.git" clone-http-non-default-hash &&
-	git -C clone-http-non-default-hash rev-parse refs/bundles/topic >actual &&
+	git -C clone-http-non-default-hash rev-parse refs/bundles/heads/topic >actual &&
 	git -C clone-from rev-parse topic >expect &&
 	test_cmp expect actual
 '
@@ -553,12 +553,12 @@ test_expect_success 'clone bundle list (HTTP, any mode)' '
 	git -C clone-any-http cat-file --batch-check <oids &&
 
 	git -C clone-list-file for-each-ref --format="%(refname)" >refs &&
-	grep "refs/bundles/" refs >actual &&
+	grep "refs/bundles/heads/" refs >actual &&
 	cat >expect <<-\EOF &&
-	refs/bundles/base
-	refs/bundles/left
-	refs/bundles/merge
-	refs/bundles/right
+	refs/bundles/heads/base
+	refs/bundles/heads/left
+	refs/bundles/heads/merge
+	refs/bundles/heads/right
 	EOF
 	test_cmp expect actual
 '
@@ -641,9 +641,9 @@ test_expect_success 'clone incomplete bundle list (http, creationToken)' '
 	test_cmp expect actual &&
 
 	# We now have only one bundle ref.
-	git -C clone-token-http for-each-ref --format="%(refname)" "refs/bundles/*" >refs &&
+	git -C clone-token-http for-each-ref --format="%(refname)" "refs/bundles/heads/*" >refs &&
 	cat >expect <<-\EOF &&
-	refs/bundles/base
+	refs/bundles/heads/base
 	EOF
 	test_cmp expect refs &&
 
@@ -679,13 +679,13 @@ test_expect_success 'clone incomplete bundle list (http, creationToken)' '
 	test_cmp expect actual &&
 
 	# We now have all bundle refs.
-	git -C clone-token-http for-each-ref --format="%(refname)" "refs/bundles/*" >refs &&
+	git -C clone-token-http for-each-ref --format="%(refname)" "refs/bundles/heads/*" >refs &&
 
 	cat >expect <<-\EOF &&
-	refs/bundles/base
-	refs/bundles/left
-	refs/bundles/merge
-	refs/bundles/right
+	refs/bundles/heads/base
+	refs/bundles/heads/left
+	refs/bundles/heads/merge
+	refs/bundles/heads/right
 	EOF
 	test_cmp expect refs
 '
@@ -721,9 +721,9 @@ test_expect_success 'http clone with bundle.heuristic creates fetch.bundleURI' '
 	test_cmp expect actual &&
 
 	# only received base ref from bundle-1
-	git -C fetch-http-4 for-each-ref --format="%(refname)" "refs/bundles/*" >refs &&
+	git -C fetch-http-4 for-each-ref --format="%(refname)" "refs/bundles/heads/*" >refs &&
 	cat >expect <<-\EOF &&
-	refs/bundles/base
+	refs/bundles/heads/base
 	EOF
 	test_cmp expect refs &&
 
@@ -749,10 +749,10 @@ test_expect_success 'http clone with bundle.heuristic creates fetch.bundleURI' '
 	test_cmp expect actual &&
 
 	# received left from bundle-2
-	git -C fetch-http-4 for-each-ref --format="%(refname)" "refs/bundles/*" >refs &&
+	git -C fetch-http-4 for-each-ref --format="%(refname)" "refs/bundles/heads/*" >refs &&
 	cat >expect <<-\EOF &&
-	refs/bundles/base
-	refs/bundles/left
+	refs/bundles/heads/base
+	refs/bundles/heads/left
 	EOF
 	test_cmp expect refs &&
 
@@ -795,12 +795,12 @@ test_expect_success 'http clone with bundle.heuristic creates fetch.bundleURI' '
 
 	# received merge ref from bundle-4, but right is missing
 	# because we did not download bundle-3.
-	git -C fetch-http-4 for-each-ref --format="%(refname)" "refs/bundles/*" >refs &&
+	git -C fetch-http-4 for-each-ref --format="%(refname)" "refs/bundles/heads/*" >refs &&
 
 	cat >expect <<-\EOF &&
-	refs/bundles/base
-	refs/bundles/left
-	refs/bundles/merge
+	refs/bundles/heads/base
+	refs/bundles/heads/left
+	refs/bundles/heads/merge
 	EOF
 	test_cmp expect refs &&
 
@@ -862,7 +862,7 @@ test_expect_success 'creationToken heuristic with failed downloads (clone)' '
 	test_cmp expect actual &&
 
 	# All bundles failed to unbundle
-	git -C download-1 for-each-ref --format="%(refname)" "refs/bundles/*" >refs &&
+	git -C download-1 for-each-ref --format="%(refname)" "refs/bundles/heads/*" >refs &&
 	test_must_be_empty refs &&
 
 	# Case 2: middle bundle does not exist, only two bundles can unbundle
@@ -909,10 +909,10 @@ test_expect_success 'creationToken heuristic with failed downloads (clone)' '
 	test_cmp expect actual &&
 
 	# bundle-1 and bundle-3 could unbundle, but bundle-4 could not
-	git -C download-2 for-each-ref --format="%(refname)" "refs/bundles/*" >refs &&
+	git -C download-2 for-each-ref --format="%(refname)" "refs/bundles/heads/*" >refs &&
 	cat >expect <<-EOF &&
-	refs/bundles/base
-	refs/bundles/right
+	refs/bundles/heads/base
+	refs/bundles/heads/right
 	EOF
 	test_cmp expect refs &&
 
@@ -961,11 +961,11 @@ test_expect_success 'creationToken heuristic with failed downloads (clone)' '
 	test_cmp expect actual &&
 
 	# fake.bundle did not unbundle, but the others did.
-	git -C download-3 for-each-ref --format="%(refname)" "refs/bundles/*" >refs &&
+	git -C download-3 for-each-ref --format="%(refname)" "refs/bundles/heads/*" >refs &&
 	cat >expect <<-EOF &&
-	refs/bundles/base
-	refs/bundles/left
-	refs/bundles/right
+	refs/bundles/heads/base
+	refs/bundles/heads/left
+	refs/bundles/heads/right
 	EOF
 	test_cmp expect refs
 '
@@ -1083,15 +1083,15 @@ test_expect_success 'creationToken heuristic with failed downloads (fetch)' '
 	test_cmp expect actual &&
 
 	# Check which bundles have unbundled by refs
-	git -C fetch-1 for-each-ref --format="%(refname)" "refs/bundles/*" >refs &&
+	git -C fetch-1 for-each-ref --format="%(refname)" "refs/bundles/heads/*" >refs &&
 	cat >expect <<-EOF &&
-	refs/bundles/base
-	refs/bundles/left
-	refs/bundles/lefter
-	refs/bundles/merge
-	refs/bundles/right
-	refs/bundles/righter
-	refs/bundles/top
+	refs/bundles/heads/base
+	refs/bundles/heads/left
+	refs/bundles/heads/lefter
+	refs/bundles/heads/merge
+	refs/bundles/heads/right
+	refs/bundles/heads/righter
+	refs/bundles/heads/top
 	EOF
 	test_cmp expect refs &&
 
@@ -1144,12 +1144,12 @@ test_expect_success 'creationToken heuristic with failed downloads (fetch)' '
 	test_cmp expect actual &&
 
 	# Check which bundles have unbundled by refs
-	git -C fetch-2 for-each-ref --format="%(refname)" "refs/bundles/*" >refs &&
+	git -C fetch-2 for-each-ref --format="%(refname)" "refs/bundles/heads/*" >refs &&
 	cat >expect <<-EOF &&
-	refs/bundles/base
-	refs/bundles/left
-	refs/bundles/merge
-	refs/bundles/right
+	refs/bundles/heads/base
+	refs/bundles/heads/left
+	refs/bundles/heads/merge
+	refs/bundles/heads/right
 	EOF
 	test_cmp expect refs &&
 
@@ -1204,13 +1204,13 @@ test_expect_success 'creationToken heuristic with failed downloads (fetch)' '
 	test_cmp expect actual &&
 
 	# Check which bundles have unbundled by refs
-	git -C fetch-3 for-each-ref --format="%(refname)" "refs/bundles/*" >refs &&
+	git -C fetch-3 for-each-ref --format="%(refname)" "refs/bundles/heads/*" >refs &&
 	cat >expect <<-EOF &&
-	refs/bundles/base
-	refs/bundles/left
-	refs/bundles/lefter
-	refs/bundles/right
-	refs/bundles/righter
+	refs/bundles/heads/base
+	refs/bundles/heads/left
+	refs/bundles/heads/lefter
+	refs/bundles/heads/right
+	refs/bundles/heads/righter
 	EOF
 	test_cmp expect refs
 '
