Just happened to me and lead me to think that the site in question is down... "Behaves misleadingly" in the summary means that such access will be loading infinitely until the auth dialog that belongs to the other Firefox window (and blocking it as it is modal) is proceeded. Thanks to being modal, this is not the case with a single window (the dialog won't let you do anything else), but the problem emerges with more windows. I would expect the tabs at the other windows (i.e., the ones that does not own the auth dialog but are blocked by the one existing at the other window) to either bring the original dialog to front (preferred) or at least display something like "HTTP authentication pending", not to present infinite page load misleading the user to think there is something wrong with the site. This should be even greater concern if auth dialog is not window-modal (current state) but rather tab-modal [1]. One could then get lost about what is going on with the infinite load even with a single window opened (common). [1] https://bugzilla.mozilla.org/show_bug.cgi?id=613785
This was with firefox-22.0-1.fc18.x86_64, but I don't expect any change towards more recent releases in this regard.
I expect that globally modal dialog (i.e., atop all FF windows) -- also a possible solution -- would clash with the mentioned heading towards tab-modal dialog.
Sidenote: having more windows opened was not desired, but a consequence of equally annoying bug [2] and extension [3] seems to be of no help; probably because I am using TreeStyleTab and bug489729 extension declared incompatibility with that, unfortunately. [2] https://bugzilla.mozilla.org/show_bug.cgi?id=489729 [3] https://addons.mozilla.org/en-us/firefox/addon/bug489729-disable-detach-and-t/versions/
Let's track it upstream.