diff --git a/builtin-describe.c b/builtin-describe.c
index d4efb10ddfddedb1671fa91130e5fe78400ad154..71be2a9364748668996696f6c74057dba43315b5 100644
--- a/builtin-describe.c
+++ b/builtin-describe.c
@@ -200,7 +200,7 @@ static void describe(const char *arg, int last_one)
 		die("%s is not a valid '%s' object", arg, commit_type);
 
 	n = cmit->util;
-	if (n) {
+	if (n && (tags || all || n->prio == 2)) {
 		/*
 		 * Exact match to an existing ref.
 		 */
diff --git a/t/t6120-describe.sh b/t/t6120-describe.sh
index c050f94bc626c340d27ebfca824116f12b4fda26..065deadc29eb3838f391ce758c4b188249dc87f9 100755
--- a/t/t6120-describe.sh
+++ b/t/t6120-describe.sh
@@ -92,12 +92,18 @@ check_describe A-* HEAD^
 check_describe D-* HEAD^^
 check_describe A-* HEAD^^2
 check_describe B HEAD^^2^
+check_describe D-* HEAD^^^
 
 check_describe c-* --tags HEAD
 check_describe c-* --tags HEAD^
 check_describe e-* --tags HEAD^^
 check_describe c-* --tags HEAD^^2
 check_describe B --tags HEAD^^2^
+check_describe e --tags HEAD^^^
+
+check_describe heads/master --all HEAD
+check_describe tags/c-* --all HEAD^
+check_describe tags/e --all HEAD^^^
 
 check_describe B-0-* --long HEAD^^2^
 check_describe A-3-* --long HEAD^^2