When you use the web browser do you get any warnings about the certificate or do get any kind of about continuing to your Tableau server ? You don't mention you do, but i had to ask.
I've had those in the past with Tableau 7, and using self signed certificates. I ended up turning off a load of IE settings in the 'Advanced' part of 'Internet Explorer' to do with Certificates. I unchecked 'Check for server certificate revocation*' Its not great that I had to do that..
Another way to verify is before blasting IE settings to bits - is to start tableau desktop, blank workbook (start from scratch) choose a dash board and Drag and drop 'Web Page' from the dashboard panel on the left into the sheet... In the 'Edit url' put your https://<your-internal-server.fqdn>/ then try the 'server->login' menu etc etc
It's no solution I know - but at least you'd be able to check if its the IE settings that is causing the issue.
When using a browser I was getting no errors/warnings at all - it was working perfectly as I would expect. I ran the Tableau Desktop with Fiddler running along side to see the web traffic and it seems that the desktop app falls over when navigating the certicate chain.
Since then I've bought a Comodo certificate (rather than the internal one I was using before) and everything, both browser and Tableau desktop, now works fine. Problem solved!
The only thing I can think of is that somewhere within the Tableau desktop application it is set to only trust certain well known certificate authorities. I've had a dig in the registry and install files but can't see where this behaviour comes from. Anyway, not a problem anymore but it would be interesting to know the reason sometime.