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If I want to describe something as "shaped like a bull's eye," would I say "bull's-eye-shaped," "bull's eye-shaped," or just stick with "shaped like a bull's eye? "

"Bull's-eye-shaped" is probably correct. "Bull's eye-shaped" would not be.
For ease of reading, I'd go with "shaped like a bull's eye" for a literal bovine eye or "shaped like a bull's-eye" for a target.

"Shaped like a bull's-eye" is the one that seems most correct to me.