Well there's always the Regia Marina able and willing to lend a hand.
But in all seriousness... if the Axis decides to go after Middle Eastern oil rather than the Baku oil, an invasion of Turkey is the last thing they'd want to do.
This would dictate a Mediterranian strategy you'd have to neutralize Malta, Gilbraltar and the Suez, pick two, before you could even think about landing troops.
A landing in Palestine thrusting inland towards Iraq would get the job done, though as I mentioned, you'd have to neutralize the British Mediterranian fleet first.
I don't think the Regia Marina is up to the task to do that on their own, even with the Luftwaffe helping them out. Their biggest weakness was that they didn't have a naval air wing and were reliant on a seperate air force branch to provide their air cover, reconissance and attacks. Coordination was bad to say the least. The RM was actually well trained, even if their ships were a bit old(the same was true of Britain though), but their tactics were definitely better.
Gibraltar would be hard without Spain joining the Axis, due to naval insufficencies.
Malta could be taken at great cost. Both Germany and Italy have servicable paratroop divisions.
The Suez... North Africa is a logistic nightmare, especially with Malta in the British hands. Though an Afika Korps reinforcing Italy, could concievably sieze the Suez.
Of course by the time the Axis do manage to acomplish Malta and the Suez, what are they going to have left for a Middle Eastern adventure?