I found myself with login prompt on SharePoint website with Fully Qualified Host Name (FQDN) host header, even though workstation and user is member of Active Directory domain. FQDN host header is for example portal.mysite.com instead of single-name host header portal.
In my case problem was that IE mistaken internet zone for portal.mysite.com as Internet, and Internet zone is configured that it prompts user with login dialog. To resolve this I simply added site to local Intranet:
Open IE and go to Tools -> Internet Options
Select Security tab and click Local Intranet zone
Click on Sites button and Click Advanced
Add your site to local Intranet zone, click Close and press OK
Click custom level and at the end choose Automatic logon with current username and password.