Lars, et al.,
I ended up using a single database, with tables for both joomla and phpbb. I also used Mehdi's "Joomla-Phbb Bridge".
The only (minor) problem was a result of not following instructions properly and trying to gzip joomla. (A no-no with the bridge and phpbb.)
Overall, I'm happy so far, although I haven't yet let my users have access to it.