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Multiplication. Incidently, there is a multiplicative version of the lemma, showing that if two countable models have the same standard system, then their multiplicative reducts are isomorphic. The proof is essentially the same overall structure as I gave above. The standard system is coded into the multiplicative structure, which is computably saturated, and from this the back-and-forth argument shows that the multiplicative structure is determined.

Conclusion: the additive cluster structure of a countable model of arithmetic also knows the multiplicative reduct of the original model up to isomorphism!

Uncountable models. Regarding uncountable models, I know only a few things, but not the full picture.

First, the lemma above is not true for uncountable models, since one can have two models of PA of size $\omega_1$ with the same standard system, but one is $\omega_1$-like and one is not, so the additive reducts are not isomorphic. So one will need other ideas.

Meanwhile, evidently it is a theorem of Harnak that all $\omega_1$-like models of $\PA$ have isomorphic additive reducts. And being $\omega_1$ like is visible in the additive cluster structure, so this provides a sufficient condition for the additive cluster structure knows the additive reduct of the original model.

Next, we argue (following Emil's comment) that the standard system is also revealed in the additive cluster structure. Namely, we can use two nonstandard elements $x<y$ to code the bits that consist of the standard part of the binary representation of the fraction $x/y$ as $M$ sees it. Note that this is visible from the additive cluster structure, since $\frac p{2^k}\leq \frac xy$ if and only if $p\cdot y\leq 2^k\cdot x$, which is equivalent to $y+\cdots+y\leq x+\cdots x$, where we take $p$ and $2^k$ many summands, respectively. For standard $p$ and $k$, this amounts to a property of the additive structure. Note, crucially, that since $x$ and $y$ are nonstandard, the standard binary bits of $x/y$ do not change when at most a finite standard change is made to $x$ or $y$. Thus, this method of coding is well defined with respect to the additive cluster structure, and so we can use the cluster addition operation. In short, the additive cluster structure knows the standard system of $M$.

Next, we seek to appeal to an additive version of the folklore result (variously attributed to Jensen and Ehrenfeucht 76, also Smorinski 81, and Wilmers) that a countable computably saturated model of $\PA$ is determined by its theory and its standard system.

Next, we argue (following Emil's comment) that the standard system is also revealed in the additive cluster structure. Namely, we can use two nonstandard elements $x<y$ to code the bits that consist of the standard part of the binary representation of the fraction $x/y$ as $M$ sees it. Note that this is visible from the additive cluster structure, since $\frac p{2^k}\leq \frac xy$ if and only if $p\cdot y\leq 2^k\cdot x$, which is equivalent to $y+\cdots+y\leq x+\cdots+ x$, where we take $p$ and $2^k$ many summands, respectively. For standard $p$ and $k$, this amounts to a property of the additive structure. Note, crucially, that since $x$ and $y$ are nonstandard, the standard binary bits of $x/y$ do not change when at most a finite standard change is made to $x$ or $y$. Thus, this method of coding is well defined with respect to the additive cluster structure, and so we can use the cluster addition operation. In short, the additive cluster structure knows the standard system of $M$.

Next, we seek to appeal to an additive version of the folklore result (variously attributed to Jensen and Ehrenfeucht 76, also Smoryński 81, and Wilmers) that a countable computably saturated model of $\PA$ is determined by its theory and its standard system.