WordPress search issue with searching html tags

You could try using several regexps to only search outside angle brackets:

function wpse159789_posts_search( $search, $query ) {
    global $wpdb;
    if ( ! preg_match( "https://wordpress.stackexchange.com/" . $wpdb->posts . '\.post_content LIKE \'%(.+)%\"https://wordpress.stackexchange.com/", $search, $matches, PREG_OFFSET_CAPTURE ) ) {
        return $search;
    }
    $search_str = stripslashes( $matches[1][0] );
    // Cater for closed angle pairs embedded in the search string.
    for ( $i = 0, $len = mb_strlen( $search_str ); $i < $len; $i++ ) {
        $q_searches[] = '(<[^>]*>)?' . preg_quote( mb_substr( $search_str, $i, 1 ) );
    }
    $q_search = implode( '', $q_searches );
    $regexs[] = '^[^<]*' . $q_search; // Before any angle bracket.
    $regexs[] = '(<[^>]*>)[^<]*' . $q_search; // After any closed angle bracket pair.
    array_unshift( $regexs, implode( ' OR ', array_fill( 0, count( $regexs ), $wpdb->posts . '.post_content RLIKE %s' ) ) );
    $search_replace = call_user_func_array( array( $wpdb, 'prepare' ), $regexs );
    $search = substr( $search, 0, $matches[0][1] ) . $search_replace . substr( $search, $matches[0][1] + strlen( $matches[0][0] ) );
    return $search;
}

I’m not sure this covers all cases, but seems to work on limited testing … (it’d be a lot simpler if MySQL regexp supported look behind/ahead).