Is there a recursively axiomatizable theory which has no independent recursive axiomatization?

Question

Does there exist a theory $T$ (in the sense of model theory) such that $T$ is recursively axiomatizable, but there is no independent recursive axiomatization of $T$? Or, does every recursively axiomatizable theory have an independent recursive axiomatization? By independent axiomatization, I mean an axiomatization where no sentence is redundant.