This question reminds me of a magical little-known theorem of Jean Pierre Ressayre that shows that every nonstandard model of $PA$ has a model of $ZF$ as a *submodel* of its Ackermann intepretation, more specifically:

**Theorem.** [Ressayre] Suppose $(M, +, \cdot)$ is a **nonstandard** model of $PA$, and $\in_{Ack}$ is the Ackermann epsilon on $M$, i.e., $a\in_{Ack}b$ iff $\mathcal{M}$ satisfies "the $a$-th digit in the binary expansion of $b$ is 1". Then for every consistent recursive extension $T$ of $ZF$ there is a subset $A$ of $M$ such that $(A,\in_{Ack})$ is a model of $T$.

**Proof Outline:** By Löwenheim-Skolem, it suffices to consider the case when $M$ is countable. Choose a nonstandard integer $k$ in $M$, and consider the submodel $M_k$ of $(M,\in_{Ack})$ consisting of sets of ordinal rank less than $k$ [as computed within $(M,\in_{Ack})$]. "Usual arguments" show that $(M,\in_{Ack})$ has a Tarskian truth-definition for $M_k$, which in turn implies that $(M_k,\in_{Ack})$ is *recursively saturated*. Since $M_k$ is also countable, $(M_k,\in_{Ack})$ must be *resplendent* [which means that it has an expansion to every recursive $\Sigma^1_1$ theory that its elementary diagram is consistent with].

Now add a new unary predicate symbol $A$ to the language ${\in}$ of set theory and consider the (recursive) theory $T^A$ consisting of sentences of the form $\phi^A$, where $\phi \in T$, and $\phi^A$ is obtained by relativizing every quantifier of $\phi$ to $A$. It is not hard to show that $T^A$ is consistent with the the elementary diagram of $(M_k,\in_{Ack})$, so by replendence the desired $A$ can be produced.

[I will be glad to add clarifications]

Ressayre's theorem appears in the following paper:

J. P. Ressayre, *Introduction aux modèles récursivement saturés*, Séminaire Général de Logique 1983–1984 (Paris, 1983–1984), 53–72, Publ. Math. Univ. Paris VII, 27, Univ. Paris VII, Paris, 1986.