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It is a stall, but one that is recoverable (for the cobra)
The body is slammed into the air as an air brake, the first part of getting the nose up is a rapid stall to prevent the aircraft just climbing and flying up.
The aim was to maintain the altitude during this, so a rapid lift of the nose, a stall of the wings so it doesn't gain a lot of altitude and also I believe an engine set up which can still work taking in enough air to give plenty of thrust to hold the altitude.

Regaining level flight is apparently from the elevators, the nose-up airbrake is somewhat stable but once you're back on the elevators, you get more drag at the back (bottom) of the wing, and it flips the nose back down again.

So- way over the angle of attack, stall the wings so you don't gain altitude, plenty of thrust so it won't sink, plenty of drag and slowing down. Then on with the elevators, adding extra drag at the bottom (still stalled, the air isn't flowing over the wings) and it'll flip the nose back down and you recover.

The MiG-29 and Su-27 can’t truly maneuver post stall. But their control surfaces and overall aerodynamics are designed to behave a certain way even at very high AoA and low airspeeds to specifically allow the maneuvers you named. I.e. while in a cobra, bringing the elevator back to neutral will cause drag at the tail of the aircraft which imparts a torque that will pitch it out of the cobra.

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There's no aerodynamic reason they shouldn't be able to do that. Fighter pilots and aerobatic pilots have been performing maneuvers that involve deliberately stalling the wing since the very early days of flight. But perhaps a description of a very simple form of stalled-wing maneuvering will help you visualize how it works.

The first thing we have to do is correct the commonly held misunderstanding that a stalled with "loses lift." That's a very poor way to describe what happens because a stalled wing still produces lift proportional to the square of airspeed, it just does it with more drag and at a higher AOA. If you doubt that, consider that a paper airplane wing is essentially stalled all the time. A better way to think of it is that stalling the wing results in a sudden shift to a lower lift to drag ratio. The wing can still produce 1 g of lift (or however much lift you want), but at much higher drag than the same wing when it is not stalled, and at higher airspeed.

Probably the easiest stalled-wing maneuver to understand is the technique used by bush pilots to minimize damage in an off-field forced landing. The pilot flies the airplane into a stall and holds it there, with aft stick, while maintaining stability with the rudder (not with the ailerons). This results in a glide with a much steeper than normal descent angle but with a low vertical speed, because the airspeed is low. (Glide angle is inversely proportional to L/D.) The pilot can then fly the airplane to a smaller clear area on the ground than they could hit with a normal glide, because of the steeper glide path. Damage on impact is minimal because of the low airspeed and low vertical speed. In fact, with a good STOL airplane, very short landings can be made on a normal runway with no damage at all. I have done this may times in a Druine Turbi, stopping in well under 200 ft from the runway threshold.

The trick with a fighter jet is to have enough control authority to usefully maneuver the jet in yaw and roll with the wing stalled without having surfaces so large that they compromise un-stalled performance. It's also helpful to have loads of low speed thrust so you don't lose more energy during that maneuver than necessary.