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This has finally stopped.
I figured this out. If you used https and someone else has used http, then you jumped protocol. You are only signed in to one of them... whatever you use. So it would appear that you were signed out (when in fact you had never been signed in to the other protocol).
Solution: All internal links, on either protocol, are changed to absolute links which keep you on your current protocol.
Additionally, I've changed the HTML for every comment already in the database to make sure that is the case, so even the historical comments get it right.
This has finally stopped. Quoting for replies still isn't working as it should for any of the browsers I use, otoh (firefox, iceweasel, android browser).