**Weak Version**:
Is there a 1st order language $L$ (with only countably-many formulas) such that for each recursive coding $C$ of the formulas of $L$, there is a theory $T$ of $L$ where

$T$ is not satisfiable,

every $C$-recursive proper subtheory of $T$ (meaning the set of $C$-codes of the sentences in the subtheory is not recursive) is satisfiable, and

every proper subtheory of $T$ that isn’t $C$-recursive is unsatisfiable?

**Strong Version**:
Is there a 1st order language $L$ (with only countably-many formulas) such that there is a theory $T$ of $L$ where

$T$ is not satisfiable,

every recursively axiomatizable proper subtheory of $T$ is satisfiable, and

every proper subtheory of $T$ that isn’t recursively axiomatizable is unsatisfiable?

Restricting to languages with only countably-many formulas excludes uncountable theories (which trivially fail to be recursively axiomatizable), but this restriction could be relaxed if we require that the theory T also be countable.