diff --git a/ci/run-build-and-tests.sh b/ci/run-build-and-tests.sh
index 28cfe730ee..de08a08d59 100755
--- a/ci/run-build-and-tests.sh
+++ b/ci/run-build-and-tests.sh
@@ -7,6 +7,12 @@
 
 export TEST_CONTRIB_TOO=yes
 
+case "$jobname" in
+almalinux-*|debian-*|fedora-*|linux-*)
+	export GIT_TEST_USE_SET_E=yes
+	;;
+esac
+
 case "$jobname" in
 fedora-breaking-changes-musl|linux-breaking-changes)
 	export WITH_BREAKING_CHANGES=YesPlease
diff --git a/t/test-lib.sh b/t/test-lib.sh
index de7d9e7b92..cded7bd693 100644
--- a/t/test-lib.sh
+++ b/t/test-lib.sh
@@ -15,6 +15,31 @@
 # You should have received a copy of the GNU General Public License
 # along with this program.  If not, see https://www.gnu.org/licenses/ .
 
+# Enable the use of errexit so that any unexpected failures will cause us to
+# abort tests, even when outside of a specific test case.
+#
+# Note that we only enable this on Bash 5 and newer, or when explicitly
+# requested by the user via `GIT_TEST_USE_SET_E=true`. This ib secause `set -e`
+# has wildly different behaviour across shells. The list of default-enabled
+# shells may be extended going forward.
+if test -z "$GIT_TEST_USE_SET_E" && test "${BASH_VERSINFO:=0}" -ge 5
+then
+	GIT_TEST_USE_SET_E=true
+fi
+
+# We cannot use `test-tool env-helper` here, as it's not yet available.
+case "${GIT_TEST_USE_SET_E:-false}" in
+1|on|true|yes)
+	set -e
+	;;
+0|off|false|no)
+	;;
+*)
+	echo "GIT_TEST_USE_SET_E requires a boolean" >&2
+	exit 1
+	;;
+esac
+
 # Test the binaries we have just built.  The tests are kept in
 # t/ subdirectory and are run in 'trash directory' subdirectory.
 if test -z "$TEST_DIRECTORY"