odd authentication issue

We just went live with a SharePoint 2013 Foundations site. There is a subsite for bidding that is anonymous at the top level which is inherited throughout all the 100 or so document libraries.

1. Users running Windows 7 and some sort of Internet Explorer can go to the site with being prompted for credentials.
2. Some Adobe documents will open without being prompted for credentials, some others do prompt. This appears to be random.
3. On same computers opening document with Firefox or Chrome open without prompt.
4. On same computers right clicking documents in Internet Explorer and saving them save the files without prompt.
5. All computers running Windows XP and Internet Explorer 8 open without prompt.
6. The only other symptom I can see is when the file is clicked to open the Adobe within the browser, the Adobe Reader seems to open and is covered by the credentials box.

Opening the docs in Windows XP successfully is obviously the change in the authentication mechanism Windows XP vs Windows 7 use.

I would appreciate some tips to further troubleshoot or perhaps a solution if it exists.

Thanks in advance.