As a pilot who spent almost two decades flying various military aircraft, almost all of it at low level, and having afterwards spent another nineteen years, mainly in UK flying non-military aircraft for my living, when flying in an AIAA there has never been a requirement to call any agency in particular.
Depending on where you fly, there are often more relevant agencies to call.
Depiction of these areas on the charts is only to advise pilots that they may see more military aircraft than elsewhere. Thirty years ago there was far more military aviation taking place, and it might have been more meaningful then, but no-one has ever "controlled" these areas.