It's possible this is because of the login setting in IE. Go to Tools --> Internet Options --> Security --> Custom Level. The last option at the bottom controls the security of how you send automatic login info. By default I believe it's set to login automatically to Intranet only. This setting is normally fine, but it's possible that it got reset to something else for you, or your server is not resolving as part of the intranet. Try playing around with this and see if the problem goes away. It might also be a good idea to enable cookies if they are disabled, since the AP does use them to store login info.
Onr last thing to check: I've noticed weird login prompt behavior like you are experiencing when testing the web server in IIS. On investigation, I realized that the problem was in QEMC --> System --> Web Servers --> Accesspoint tab. If you put anything into the "Login address" field, you will get a login prompt even if authentication is set to "Always." Therefore, delete the contents of the "Login address" field, restart IIS, and clear IE temporary files. Hopefully this will resolve the problem.