I had an “interesting” problem on an Exchange server recently… I could access the Exchange Control Panel using https://localhost/ecp but it I tried to use its externally-accessible namespace (which I needed to in order to run the Exchange Hybrid Configuration Wizard), it kept looping on the authentication prompt. The password was correct but it just came straight back to a prompt again…
It turned out to be an issue with HTTP redirection in IIS and, following the MSExchangeGuru post on HTTP redirection issues with the ECP virtual directory I set up HTTP redirection on the Default Website but turned it off for OWA, ECP and the various other virtual directories. I also checked that SSL was not required for the PowerShell virtual directory.
Bipin Giri’s post on configuring URL redirection in Exchange 2013 suggests I need to repeat the same steps for the Exchange Backend Server website too but I didn’t find that necessary (and I’m sure I’ve also read to leave that alone!).
There’s a useful table in the MSExchangeGuru post that summarises the need for an HTTP redirect and whether SSL should be enabled for the website and for each virtual directory but it’s also worth knowing that Morten Nielsen has listed the default virtual directory settings for Exchange 2013 in great detail.