sorry about the bad info in my first post - i did not know that you had an alarm in there before. in addition, someone may have wired the radio up much like the RAP (retained accessory power) circuit on 4th gen camaros. with that, the radio, power locks, etc stay functional for 10 mins or until a door is opened.
the most common way i know of that people wire the radio like that (any car) - all it takes is a latching relay, most commonly made from 2 bosch style SPDT relays. one relay runs the door pin through a coil. the same relay has ground output through the NC contacts (30 to 87A). this feeds to the second relay. the second relay has this switched ground from the first relay going through a coil terminal.
so the first relay functions normally through the normally closed circuit. this feeds ground to the second relay to energize it. but when a door is opened and kills the first relay (switches off the output from the previously closed 87A), this removes the ground from the coil of the second relay, and the radio kills.
i know that was hard to follow. i will see if i have a pic of the latching relays if you want, not that it matters really.
do you hear any relay clicking when you turn on the radio (with the set up given, the second relay needs to energize to activate power to the radio). when you open the door with the car off, the door trigger energizes the first relay, so you may hear that too.
worst case if you cannot find the relays (if that sounds like the problem), i would run a new key on power wire from the fuse box to the radio. that would by pass whatever someone else did.
my best guesses. good luck.