Complete one-consistent theories can prove false sentences

Given a bounded-complete one-consistent arithmetic first-order theory with a false theorem, enumerate first-order sentences and successively choose one-consistent one-sentence extensions. Every stage has Sigma-1 soundness, and a false Sigma-1 theorem in the union would already have a formal proof at one stage. The union is therefore one-consistent and complete while retaining the false theorem. It cannot be omega-consistent, since a complete bounded-complete omega-consistent first-order theory is sound.