Here you go:
CREATE FUNCTION `strip_tags`($str text) RETURNS text
BEGIN
DECLARE $start, $end INT DEFAULT 1;
LOOP
SET $start = LOCATE("<", $str, $start);
IF (!$start) THEN RETURN $str; END IF;
SET $end = LOCATE(">", $str, $start);
IF (!$end) THEN SET $end = $start; END IF;
SET $str = INSERT($str, $start, $end - $start + 1, "");
END LOOP;
END;
I made sure it removes mismatched opening brackets because they're dangerous, though it ignores any unpaired closing brackets because they're harmless.
mysql> select strip_tags('hello wo<>rld <again<.');
+----------------------------------------------------------------------+
| strip_tags('hello wo<>rld <again<.') |
+----------------------------------------------------------------------+
| hello world again. |
+----------------------------------------------------------------------+
1 row in set