I have a phorum forum with approx 15,000 posts. Phorum is good. But I want a forum that can appear as the contents of a page division inside my CMS, and not as separate pages, outside my own layout. Phorum (and most other forums) cannot do that. Not outside the iframe context anyway. And I do not want to use an iframe. Is there any such forum? ...that can become a plugin module to any arbitrary CMS (perhaps with the aid of some wrapper package I would have to write myself).
I think Joomla supports iframe by default, so integration would be easier there. Same for Drupal, that has a module for Phorum.
Also note that, with some limited skills, it should be possible to create a Phorum theme that will look like your website layout, giving the feeling both of them are merged.
Concerning technical integration, the most important is to know what CMS you have, in order to see if there is a forum software that could be directly integrated with it.
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