we failed to find any "plug-and-play" solution for this, supposingly widespread, problem. So what's the right way to approach it?
Assuming the forum is on a subdomain: pass a unique authentication token from your main site to your forum to authenticate users when they visit the forum.
The most fashionable way to accomplish this is by using JSON Web Tokens (JWT) which you read about here: http://jwt.io/ in conjunction with a javascript callback from the forum to the main site to check the user's status (e.g. is the user logged in to the main site).
Some hosted forum solutions support SSO right out of the box or with minimal work. Here are two examples: