We are experiencing login problems when people are switching between the in-house URL and the alternate full URL. For instance, if one's browsing http://intranet and then typing in http://intranet.company.com in the browser, you will get a new login request, which will most likely not go away until you click 'Cancel' and then refresh the page.
Another problem is that if someone is attaching pictures to a list item (the pictures are shown on the front page) by using one of the URL's, people browsing with the other URL will receive access denied for these images, and thereby will get a login prompt per image. After 'Canceling' these prompts (it won't work to login), the images will not show.
Both intranet and intranet.company.com are added as AAMings. I wonder if the problem could be related to the DNS-configuration, but I really don't know. We're using Claims with only FBA.
I can add that when I'm on the http://intranet and asks to log in as differnet user, the login prompt asks me to connect to http://intranetsrv01.domain, while when I'm on http://intranet.company.com and tries the same, it asks to log in to http://intranet.company.com
Update: I'm pretty sure the AAM are configured errorenously. Please take a look at this:
Internal URL Zone Public URL for Zone http://intranet.company.com Default http://intranet.company.com http://intranet Intranet http://intranet http://intranetsrv01 Custom http://intranetsrv01