For $\mathcal{M}$ a (countable) nonstandard model of $\mathsf{PA}$, let $\mathsf{SS}(\mathcal{M})$ be the set of sets of natural numbers coded by elements of $\mathcal{M}$. There are various ways to define this, for example $$\{X\subseteq\mathbb{N}:\exists a\in\mathcal{M}\;\forall k\in\mathbb{N}\;(k\in X\iff p_k\vert a)\}$$ (where $p_k$ is the $k$th prime and we conflate $\mathbb{N}$ with its canonical image in $\mathcal{M}$).

I'm curious about how this could differ from the following analogue: let $$\mathsf{SS}^-(\mathcal{M})=\{X\subseteq\mathbb{N}:\exists a\in\mathcal{M}\;\forall k\in\mathbb{N}\;[k\in X\iff \exists n\in\mathbb{N}\;(p_k^n\not\vert a)]\}.$$

Intuitively, elements of $\mathbb{N}$ are **prevented** from entering $X$ by corresponding primes dividing $a$ "too much." (This version has come up in a separate problem I'm playing with, and I'd like to understand it better.)

It's easy to show that $\mathsf{SS}(\mathcal{M})$ is the set of elements of $\mathsf{SS}^-(\mathcal{M})$ whose complements are also in $\mathsf{SS}^-(\mathcal{M})$, and it's not hard to show that every $\mathcal{M}$ has an elementary extension $\mathcal{N}$ such that $\mathsf{SS}^-(\mathcal{N})=\mathsf{SS}(\mathcal{N})$. However, this still leaves a lot open. In particular:

Is there an $\mathcal{M}$ with $\mathsf{SS}(\mathcal{M})\not=\mathsf{SS}^-(\mathcal{M})$?

Equivalently per the above, is there an $\mathcal{M}$ whose $\mathsf{SS}^-$ is not closed under complementation?