None of the summoned primals are "real" - they are what those who summon wish them to be. They really are the "eikon" - false gods - that the Garleons purport them to be. The Twelve deities that guard Eorzea could, in theory, be summoned as well.

The Ala Mihgan youths foolishly attempted to summon Rhalgr as a primal. I'm frankly surprised the Ishgardians didn't at some point try to summon Halone.

Edit: Not enough coffee yet.