Annoyingly, I am not able to replicate what I saw earlier. I am now keeping the two tabs open and am monitoring them. For now they are remaining in sync. When I archive a message in one tab, the other tab shows the “See 1 new or updated topic” blue notice at the top of the list. Selecting that brings it back in sync.
I do notice the blue “new” notice appears next to messages that I haven’t seen, specific to that member.
So this is all making more sense now. I will keep an eye on it.