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I tried the solution of Onetrickpony above https://wordpress.stackexchange.com/a/5404/37612, which is great, but I found one issue there, which did not work for me, and I would make one small modification:

1. if I searched for a string in the title of the taxonomy - it works great

2. if the taxonomy has special characters e.g. with german "Umlauts" (ö,ä,ü) and one searches for oe, ae, ue insteda of using the special char - you need to add the search in the slug of the taxonomy - OR t.slug LIKE '%".get_search_query()."%'

3. if you search for a combination of a search query and a taxonomy filter - this also works fine

4. But the problem is, when you try to use only the taxonomy filter - the search hook append an empty string to the query if no text is searched for, and for that reason you get ALL posts in the result, instead of only those from the filtered taxonomy. A simple IF statement solves the problem. So the whole modified code would be this (works perfectly fine for me!)

function custom_search_where($where){ 
  global $wpdb;
  if (is_search() && get_search_query())
    $where .= "OR ((t.name LIKE '%".get_search_query()."%' OR t.slug LIKE '%".get_search_query()."%') AND {$wpdb->posts}.post_status = 'publish')";
  return $where;
}

function custom_search_join($join){
  global $wpdb;
  if (is_search()&& get_search_query())
    $join .= "LEFT JOIN {$wpdb->term_relationships} tr ON {$wpdb->posts}.ID = tr.object_id INNER JOIN {$wpdb->term_taxonomy} tt ON tt.term_taxonomy_id=tr.term_taxonomy_id INNER JOIN {$wpdb->terms} t ON t.term_id = tt.term_id";
  return $join;
}

function custom_search_groupby($groupby){
  global $wpdb;

  // we need to group on post ID
  $groupby_id = "{$wpdb->posts}.ID";
  if(!is_search() || strpos($groupby, $groupby_id) !== false || !get_search_query()) return $groupby;

  // groupby was empty, use ours
  if(!strlen(trim($groupby))) return $groupby_id;

  // wasn't empty, append ours
  return $groupby.", ".$groupby_id;
}

add_filter('posts_where','custom_search_where');
add_filter('posts_join', 'custom_search_join');
add_filter('posts_groupby', 'custom_search_groupby');