Luckily not much was changed so it was pretty easy to track it down.
In 1.1.19 they added some code in LogInOut.php that changes the users password salt whenever you log out. When you switch to another browser, the hash matching fails since it was generated with a different salt, and you get logged out.
The offending code is at lines 463-464. I have no idea how to make a modification for SMF but I guess you could just try commenting out those lines to start with, to see if it works.
The call to session_destroy() on line 462 is also added in that change, but I would expect that is independent for each browser session so I don’t think that’s causing any issues.