[_ck_]

You have the right approach if you find that log-outs are out of sync.

You need to watch the cookie names and paths.

MAKE SURE you clear ALL cookies while you are experimenting with paths and names. If you leave left overs, you are going to get strange behaviors.

If you leave the page open, even after clearing cookies you may have form post data or session data in there. Close page, clear cookies, THEN confirm behavior.

If you find that path names and cookie names are absolutely identical, it’s possible that the logout process for WPMU is not robust enough in the way it resets cookies. ie. doesn’t set the date far enough in the past.

Following the WPMU tag will probably help you bunches. ie. https://bbpress.org/forums/topic/cookie-hell-wpmu-bbpress-bloodshot-eye-balls