If intense shockwaves form on the upper surfaces of the wings, the resulting boundary separation will reduce the lift being generated and also envelope the rear surface of the aircraft (including the tail plane) in turbulent airflow. This will directly reduce the ability of the elevators to generate control forces.
Australian built single seat Mk 30 Vampires had this problem in the early 1950's. To accommodate the larger airflow requirements of the RR Nene engine, a pair of intakes were installed on the top of the fuselage directly aft of the cockpit. These intakes had a curved top surface something like a typical carburettor air intake seen on radial engines like the DC3, DC4, Beaufighter and so on. It was discovered (too late to save three crashes in which the pilots were killed because in those days the Vampires did not have ejection seats) that at around Mach 0.68 in a dive, shock waves formed on the top of these two intakes and caused loss of elevator effectiveness which could not be regained until lower altitudes. The intakes were known as "Elephant Ears"
For example, two Vampires experienced what was known then as compressibility while practicing formation aerobatics around 15,000 ft. Once over the top during a loop the pair would deploy airbrakes to keep speed under control in the dive recovery. In the accident case the leader did not deploy his air brakes in the dive and was unable to pull out from the dive due to loss of elevator effectiveness. The No 2 aircraft formating closely was similarly affected by compressibility and both went in.
It wasn't until the Australian test pilot for de Havillands at Bankstown, NSW, "Blackjack" Walker was tasked to investigate the behaviour of the Vampire at the Critical Mach Number, that compressibility was found to be the problem. If I recall, he took a Nene powered (no ejection seat) Vampire to 40,000 ft and rolled into a steep dive. He described losing elevator control and was only able to recover by 10,000 ft. It was a mighty close shave. The fix was to relocate the elephant ear intakes under the fuselage and the problem disappeared but not before another fatal crash due to compressibility. That aircraft was practicing a diving quarter attack at another Vampire around 20,000 ft when the pilot reported that he was in compressibility. The aircraft was seen to crash almost vertically.
When this writer first flew the single seat Vampire Mk 30 (no dual available in those early days and all of us on the fighter course only had about 250 hours total time in our log books) we would accompany a experienced Vampire pilot in formation for what was called Mach Runs. He would stay very close formation as No 2 and talk us into the dive. He knew from experience the symptoms of compressibility before we would be aware of it.
As soon as it was felt and the elevators were becoming ineffective he would call on the radio for throttle close and dive brakes out before things got out of hand in the dive. Eventually all the single seat Vampires in Australia were modified to have the intakes installed under the fuselage and as an added bonus, ejection seats were installed.