For a nonstandard model of enough arithmetic - say, $\mathcal{N}\models I\Sigma_1$ - we can define the set of *halting times of standard machines* relative to $\mathcal{N}$: $$SH(\mathcal{N})=\{n\in\mathcal{N}:\exists e\in\omega(\mathcal{N}\models \text{$\Phi_e(0)$ halts in exactly $n$ steps})\}.$$ Here $\omega$ denotes standard $\omega$, so $SH(\mathcal{N})$ is not in general definable in $\mathcal{N}$. Note that $SH(\mathcal{N})$ is a subsemiring of $\mathcal{N}$, so we can consider it as a possible setting for arithmetic.

Sadly, $SH(\mathcal{N})$ is *not* guaranteed to be a model of $I\Sigma_1$, regardless of how much induction we have in $\mathcal{N}$. To see why: Suppose we have a $\Sigma_0$-formula $\varphi$ such that $SH(\mathcal{N})\models \exists x\varphi(x).$ Then certainly $\mathcal{N}\models\exists x\varphi(x)$, so there is a least $m\in\mathcal{N}$ such that $\mathcal{N}\models\varphi(m)$; however, this doesn't give us a least element of $SH(\mathcal{N})$ satisfying $\varphi$.

I'm in general interested in the arithmetic properties of $SH(\mathcal{N})$. In the interests of asking a concrete question, however:

Are there nonstandard models $\mathcal{N}$ of $I\Sigma_1$ with $SH(\mathcal{N})\models I\Sigma_1$? If so, what determines whether $SH(\mathcal{N})\models I\Sigma_1$?