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Of course, people DID build ternary computers.

The most notable of these is Setun' (I had a chance to play with it at Moscow State University).

Setun is the only example, and it had a binary memory.

There was also the Canadian QTC-1. [0]

This is ROM only, not a fully functioning computer.

Head to the end of the paper about the ROM [0], and you'll find the QTC-1 was the computer said ROM was designed for, not the ROM design itself.

AFAIK it was never built, only partially designed.