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What was the prototype the Soviets displayed at Tushino in 1956 and that NATO named instantly?
Short answer
The prototype the Soviets displayed at Tushino for Aviation Day 1956 was taken for a new interceptor, which NATO immediately named Faceplate; it was the MiG-21. In the West, the lack of information led to its early details being confused with those of other Soviet fighters.

The Aviation Day prototype
At their last mass military aviation display at Moscow's Tushino airfield, on the occasion of Aviation Day in 1956, the Soviets presented the prototype of a new interceptor. That aircraft was the MiG-21, and NATO promptly assigned it the code name Faceplate. The first prototype to carry the delta wing of the production variants was the Ye-4. This prototype had made its first flight on 16 June 1955, so the Tushino appearance of July 1956 was its first public showing.
A specimen assembled in a hurry
The aircraft on display had been put together rather hastily for the show, and test flights both before and after revealed numerous shortcomings. The Soviets worked quickly: by October 1957 they already had a more modern version of the same aircraft in series production.
The picture that reached the West
In the West, the scarcity of available information meant that the first details of the MiG-21 were often confused with those of similar Soviet fighters of those years. In the variant nomenclature, the letter P stands for Perekhvatchik, "interceptor", and F for Forsirovannyy, "uprated".
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