I have an AudioControl OEM integration product installed and my aftermarket amps stay on after the radio is turned off.

There is more to the factory audio system than the radio. If the OEM system is on then the aftermarket pieces will be on too. Often, new cars "wake up" when you use the remote to unlock the doors. This turns on the factory amp so when the door opens, you hear the chime. When you turn off and exit the car, the amp waits a little longer before turning off completely, this too will keep your AudioControl OEM product (the LC6i or DQL-8, for instance) on until the factory amp turns off. Just wait a bit more for the car to complete its power down sequence.