One thing we could do is create a function that duplicates user from site_user_db to the forum database when a registered member from your site clicks on the forum link. Only if it wasn't duplicated already.
if site_user_db doesnt exist in forum_user_database, then insert user and login using site_user_db.
select * from site_member
insert * crucial data from site_user_db to phpbb_users (user, email, password,avatar,role, etc etc).
Then when logging in make the 2 cookies for phpbb and for the site.
Me and my friend successfully did this with phpbb2 long time ago. We'll be working on it and when we are finish we'll share it on the site. but I'm hoping there is a best solution out there, I'll be doing some research.
I think the the main challenges will be:
How to insert the password in the same format (the site's password and phpbb's password).
How's a phpbb3 password generated? I can tell it's not just a simple md5 anymore. I didn't check the files yet.
How to login 2 times (forum and site) with one form located anywhere on the site.