Is that thing an air breather?
Yep. Rocket launch with solid fuel ram-jet sustainer, IIRC. Also called a “Throttleable Ducted Rocket.”
The proposed AIM-54 Phoenix replacement, the AIM-152, was to have a similar propulsion. It was also supposed to have greater range and lighter weight than the AIM-54, however, it was also designed specifically to be launched from the F-14 platform. With the “end” of the Cold War, the perception of attack by a regiment sized wave of Backfires and Badgers was gone. Note that I heavily qualified the last sentence.