So if an offshore company is being run by a PE in Georgia, but doesn't have any Georgian sourced income then it looks like Georgia won't tax it anyway?
Looks like that's the case:
Non-resident enterprises carrying out economic activities in Georgia through a permanent establishment (PE) are subject to CIT with respect to its Georgian-source income. (source: http://taxsummaries.pwc.com/ID/Georgia-Corporate-Taxes-on-corporate-income)