From what I have seen, I believe that both types of aircrafts are controlled by the same ATC. Otherwise, the two ATCs will have to coordinate among themselves to create non intersecting vectors for each aircraft - which sounds needlessly tedious. But the military does have its own ATCs in many areas. They usually separate the airspace into 'civilian corridors' and 'military corridors', each under the control of the respective ATCs. In conflict zones though, the military operates entirely under their own control. The concept of ATC becomes a bit blurred here. The aircrafts are vectored either by ship-based controllers or more frequently by AEW&CS aircrafts. This is what I assume was happening with the tanker in the story. They were near Venezuela and the Curacao ATC had no information about the tanker.