Caching works incorrectly

Problem/Motivation

When the anonymous page cache is enabled (or Memcache module is enabled and $conf['page_cache_without_database'] = TRUE; in settings.php), a message isn't showing for anonymous users. The reason for it is /conditional_message_data_output returns nothing. That's is because the page is taken from the cache entirely.

And what do we put into the page cache?

$url = url('conditional_message_data_output', array('absolute' => TRUE));
cache_set($url, $data, 'cache_page');

It's an array instead of rendered content. And even if we render it, we still need to set the proper headers (eg content-type), right now the defaults are returned.