As far as I can tell my Exchange 2010 Mailbox and Client Access server is working fine except for Outlook Anywhere.
I fired up the Exchange Connectivity Tester and ran it against my server and I get this report:
Testing HTTP Authentication Methods for URL https://mail.contoso.com/rpc/rpcproxy.dll?server6.corp.contoso.com:6002.
The HTTP authentication test failed.
Additional details: A Web exception occurred because an HTTP 503 - ServiceUnavailable response was received from Unknown.
When I do a search for "ServiceUnavailable response was received from Unknown." I get only a couple of relevant results, including a 22k-view Exchange Forum thread, but none of the solutions discussed help.
There is nothing of relevance in the server's Event Log.
mail.contoso.com is the public domain name of the CAS/MB/HT server.
server6.corp.contoso.com is the internal domain name of the server.