Let $ T $ be a recursively axiomatizable extension of $ \mathsf {PA} $ and $ \mathfrak M $ be a model of $ T + \operatorname {Con} ( T ) $. Is it true that there must exist a model $ \mathfrak N $ such that $ \mathfrak M \subset \mathfrak N $ and $ \mathfrak N \models T + \neg \operatorname {Con} ( T ) $?
My intuition is that the answer should be affirmitive. I think since $ \neg \operatorname {Con} ( T ) $ is $ \Sigma _ 1 $, we may be able add a new element to $ | \mathfrak M | $ coding a (nonstandard) proof of contradiction in $ T $. I suppose some sort of argument similar to that of Gödel's incompleteness theorems might suffice for proving the existence of such extension; but I don't know if that's true.
What I tried to do was taking the positive diagram $ \Delta $ of $ \mathfrak M $, and trying to prove that $ T \cup \Delta + \neg \operatorname {Con} ( T ) $ is consistent. As my first attempt, I tried to find theorems from model theory, like Robinson's joint consistency theorem, that would do the job. After failing to find such theorems, I tried to prove the consistency of the mentioned theory via an argument by contradiction: if it's not consistent, finitely many members of $ \Delta $, say $ \delta _ 0 , \dots , \delta _ { n - 1 } $, would suffice to derive a contradiction. Letting $ \delta = \bigwedge _ { i < n } \delta _ i $, we would then have $ T + \delta \vdash \operatorname {Con} ( T ) $. As the parameters from $ \mathfrak M $ do not appear in $ T $ or $ \operatorname {Con} ( T ) $, we can replace the parameters appearing in $ \delta $ with variables and take the existential closure of the resulting formula to get a $ \Sigma _ 1 $ sentence $ \gamma $ in the language of arithmetic such that $ T + \gamma \vdash \operatorname {Con} ( T ) $. Now, since $ \operatorname {Con} ( T ) $ is $ \Pi _ 1 $, I doubt that the existence of such $ \gamma $ would be possible, but I couldn't go further and prove this fact.