[Tildes](https://tildes.net) is open source and does something like that. Not a direct 1:1 to phpBB but more like a cross between old school forums and reddit.
Tildes is a popular-kids-club-invite-only thing and therefore not a replacement for phpBB or Reddit. (Before you say it's open source and you should just run your own copy—they provide no instructions for setting up your own instance. It's apparently not meant to be used by anyone else.)
