I'm one of those oddballs that uses an adblocker but whitelists domains that I trust. I want you guys to get advertising revenue from me because I support the website. But, I noticed the problem resolved on my phone whenever I got around the popup ad. So, I re-enabled ad-blocking on this domain. Guess what? Problem solved. No more login issues.
Might want to look at your advertisements and see what's preventing users from remaining logged in. I could login and navigate to the subforums, but then if I clicked the forum button at the top, I'd be logged out again until I navigated elsewhere. There's an advertisement doing this.