I'd check the grounds on the tach and radio --and also check dash ground to body and firewall to engine. Sounds like you might be getting voltage back-feeding into the radio that should be completing its circuit to ground.
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Mick Batson
1967 original owner Tyro Blue/black top 4-speed HO GTO with all the original parts stored safely away -- 1965 Catalina Safari Wagon.
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