No, and the reason is the compass is simulating your basic sense of direction most people get when they have been in an area long enough. It is shown because you can't get that from sitting on your ass in a chair in front of a pc screen.
Additionally when you press tab to open your map your company issued cell phone is pulled up infront of you. If you are in a dark environment it will even light up, potentially giving away your location.
I wish that there was a similar mechanic when you opened the tac map, but I'm sure there truly would be a great wailing and nashing of teeth if that were to happen.