You know, it's kinda' funny. We contemplated rear firing missiles back in the '80's when it looked like we were going to go toe-to-toe with the Warsaw Pact. I'm curious that nobody from Boeing or Lockheed has anything to add to this thread.
The missiles were intended to be pointed aft on some "future combat aircraft" but I don't know what they intended to target the missiles. I seem to remember the missile itself had thrust vectoring.
This all became academic because nowadays most planes fly around with their radars off, waiting for AWACS to vector them to a target. If they have to fire up their radar at all, it is only to illuminate their targets.
Why worry about rear-firing missiles when we haven't had a real dogfight since vietnam?
(I blame American complacency. It would be little trouble to mount a simple thermal sight, or utilize the IR threat warning reciever pointed aft from the tail to target such a rear-firing missile.) The missile tech to maintain stability in a rearward boost would be the biggest hurdle.
It's all feasable, just not something anyone will push until we need it.
But I'm sure those Vietnam F-4 pilots would have surely welcomed such a trick against the pesky Mig-17s and Mig-19s they ended up in knife fights against.