Certainly the leadership in theatre is by the fleet commander, who is an American officer based ashore in Bahrain. He's the maritime component commander for American forces. He has a variety of international officers on staff, and some officers from both regional and western nations as subordinate commanders.
It really is as straightforward as knowing what the limits of rules of engagement are and then asking ships to do those things their governments have approved. There's really pre-approval by the government. If the Ottawa is asked to do something that is not already approved, the commanding officer will instantly get back to General Gauthier, in the face of a surprise arising in tasking, and seek direction from Canadian authorities.