On the doctor's home territory, though, that would be tricky; they might easily have a half-dozen nurses standing by with tranquilizers ready to knock out any potential interferers.

This would eventually lead to medical force escalation, and then medical feudalism. You go to your "own" hospital knowing for sure that the doctors there will not take out your organs without permission, since you're part of their tribe. However, that hospital has to put in place a strong defensive perimeter to stop any task forces from the hospital tribe down the street from breaking in and stealing organs for themselves.

And of course, if you turn out to need an organ, then your hospital would deploy its own tribal ninja doctors to sneak into the enemy hospital and retrieve whatever is needed...