TFS 2013 web access - can't sign out

We use TFS 2013 on premise. I just noticed a weird issue with the web interface, namely I can properly sign out only from Firefox.

When I use Chrome, IE or Opera, hitting the "sign out" link shows the proper signout page, saying:

Sign out: close browser to complete sign out
You must close your browser to complete the sign out process.

But after I open a new instance and hit http://servername:8080/tfs - hoping to see the "Authentication required" popup - the welcome page loads as if I had logged in already.

I erased all browsing history, temporary files, cookies and website data, form data, saved passwords, etc. for all browsers, checked Microsoft Credential Manager in case it has a TFS related entry, I rebooted the PC, but no result.

The installed browsers:
Firefox 36.0
Chrome 40.0.2214.115 m
IE 11.0.9600.17633
Opera 27.0.1689.76

This is probably caused by automatic Windows authentication with NTLM, which is caused by Internet Explorer settings.

WebKit based browsers (Chrome, new Opera) follows IE settings.

As a workaround, you can add your TFS server to "Trusted zones" and specify a custom logon action: