Our site is completely private and all users must be logged in the use anything.
Initial visits to the site prompt for a login as expected. If you open a new tab and access the same site or you close the browser without logging out it will prompt you to log in but never allow you to log back in you are in a login prompt loop.
But, if you click the link to return to the site at the bottom of the login prompt you get the home page properly.
Usually, this will not happen if you log out of the site instead of closing the page. However, I have had it occur when I signed into the site at a computer for the very first time.
This occurs on all subsites and the main site. It is a real issue trying to get people to understand they can just click the link if they are "stuck".