diff --git a/t/README b/t/README
index c730a70770..d9f65bfa6b 100644
--- a/t/README
+++ b/t/README
@@ -439,6 +439,12 @@ GIT_TEST_DEFAULT_HASH=<hash-algo> specifies which hash algorithm to
 use in the test scripts. Recognized values for <hash-algo> are "sha1"
 and "sha256".
 
+GIT_TEST_PIPEFAIL=<boolean>, when true, run 'set -o pipefail' to catch
+failures in commands that aren't the last in a pipe. Defaults to true
+on bash versions which know how to ignore SIGPIPE failures under the
+'set -o pipefail' mode (as of 2021-01-14 only in an out-of-tree patch
+to bash).
+
 Naming Tests
 ------------
 
diff --git a/t/t0000-basic.sh b/t/t0000-basic.sh
index 930cf9d1b7..e70cc37139 100755
--- a/t/t0000-basic.sh
+++ b/t/t0000-basic.sh
@@ -1350,4 +1350,18 @@ test_expect_success 'test_{must,might}_fail accept non-git on "sigpipe"' '
 	test_cmp badobjects out
 '
 
+test_expect_failure BASH_SET_O_PIPEFAIL 'test_{must,might}_fail ok=sigpipe under bash "set -o pipefail"' '
+	grep string </dev/null | true
+'
+
+test_expect_failure BASH_SET_O_PIPEFAIL 'test_{must,might}_fail ok=sigpipe under bash "set -o pipefail"' '
+	test_must_fail grep string </dev/null | true &&
+	test_might_fail grep string </dev/null | true
+'
+
+test_expect_success BASH_SET_O_PIPEFAIL 'test_{must,might}_fail ok=sigpipe under bash "set -o pipefail"' '
+	test_must_fail ok=sigpipe grep string </dev/null | true &&
+	test_might_fail ok=sigpipe grep string </dev/null | true
+'
+
 test_done
diff --git a/t/t0005-signals.sh b/t/t0005-signals.sh
index 4c214bd11c..cc5784a274 100755
--- a/t/t0005-signals.sh
+++ b/t/t0005-signals.sh
@@ -40,12 +40,12 @@ test_expect_success 'create blob' '
 	git add file
 '
 
-test_expect_success !MINGW 'a constipated git dies with SIGPIPE' '
+test_expect_success !MINGW,!BASH_SET_O_PIPEFAIL 'a constipated git dies with SIGPIPE' '
 	OUT=$( ((large_git; echo $? 1>&3) | :) 3>&1 ) &&
 	test_match_signal 13 "$OUT"
 '
 
-test_expect_success !MINGW 'a constipated git dies with SIGPIPE even if parent ignores it' '
+test_expect_success !MINGW,!BASH_SET_O_PIPEFAIL 'a constipated git dies with SIGPIPE even if parent ignores it' '
 	OUT=$( ((trap "" PIPE; large_git; echo $? 1>&3) | :) 3>&1 ) &&
 	test_match_signal 13 "$OUT"
 '
diff --git a/t/t5000-tar-tree.sh b/t/t5000-tar-tree.sh
index 3ebb0d3b65..3adcbce84c 100755
--- a/t/t5000-tar-tree.sh
+++ b/t/t5000-tar-tree.sh
@@ -416,7 +416,7 @@ test_expect_success LONG_IS_64BIT 'set up repository with huge blob' '
 
 # We expect git to die with SIGPIPE here (otherwise we
 # would generate the whole 64GB).
-test_expect_success LONG_IS_64BIT 'generate tar with huge size' '
+test_expect_success LONG_IS_64BIT,!BASH_SET_O_PIPEFAIL 'generate tar with huge size' '
 	{
 		git archive HEAD
 		echo $? >exit-code
diff --git a/t/t9902-completion.sh b/t/t9902-completion.sh
index a1c4f1f6d4..3414ac56f4 100755
--- a/t/t9902-completion.sh
+++ b/t/t9902-completion.sh
@@ -7,6 +7,11 @@ test_description='test bash completion'
 
 . ./lib-bash.sh
 
+if test -n "$GIT_TEST_PIPEFAIL_TRUE"
+then
+	set +o pipefail
+fi
+
 complete ()
 {
 	# do nothing
diff --git a/t/test-lib.sh b/t/test-lib.sh
index 9fa7c1d0f6..118dc80ffc 100644
--- a/t/test-lib.sh
+++ b/t/test-lib.sh
@@ -36,6 +36,31 @@ then
 fi
 GIT_BUILD_DIR="$TEST_DIRECTORY"/..
 
+# Does "set -o pipefail" on this bash version handle SIGPIPE? Use it!
+. "$TEST_DIRECTORY/lib-bash-detection.sh"
+GIT_TEST_PIPEFAIL_TRUE=
+GIT_TEST_PIPEFAIL_DEFAULT=false
+if test -n "$TEST_SH_IS_BIN_BASH" &&
+       $BASH -c 'set -eo pipefail; yes | head -n 1 >/dev/null'
+then
+	GIT_TEST_PIPEFAIL_DEFAULT=true
+fi
+# We're too early for test_bool_env
+if git env--helper --type=bool --default="$GIT_TEST_PIPEFAIL_DEFAULT" \
+       --exit-code GIT_TEST_PIPEFAIL
+then
+	set -o pipefail
+
+	# Only "set -o pipefail" in the main test scripts, not any
+	# sub-programs we spawn.
+	GIT_TEST_PIPEFAIL=
+	export GIT_TEST_PIPEFAIL
+
+	# For the convenience of the prereq for it.
+	GIT_TEST_PIPEFAIL_TRUE=true
+	export GIT_TEST_PIPEFAIL_TRUE
+fi
+
 # If we were built with ASAN, it may complain about leaks
 # of program-lifetime variables. Disable it by default to lower
 # the noise level. This needs to happen at the start of the script,
@@ -1552,6 +1577,10 @@ test_lazy_prereq PIPE '
 	rm -f testfifo && mkfifo testfifo
 '
 
+test_lazy_prereq BASH_SET_O_PIPEFAIL '
+	test -n "$GIT_TEST_PIPEFAIL_TRUE"
+'
+
 test_lazy_prereq SYMLINKS '
 	# test whether the filesystem supports symbolic links
 	ln -s x y && test -h y
