Exactly. I took part in NATO exercises back in the 70’s, and it doesn’t seem far fetched that some NATO person could have been in charge in some way, shape or form.
NATO in charge of a carrier though? That might be different.
Its a play nice together game. The CO still owns his ship and the airwing. But its like chopping from EURCOM to CENTCOM and back. Just defines whose command structure you report to.