This question is motivated both by the notion of zero-knowledge proofs and by general curiosity about versions of the infinitely-long Ehrenfeucht-Fraisse game which don't trivialize (= Duplicator win iff computably isomorphic) in the computable setting.
Suppose we have a pair of classically-isomorphic relational computable structures $\mathcal{A},\mathcal{B}$, each with domain $\omega$. The scrambled EF-game between $\mathcal{A}$ and $\mathcal{B}$ is played as follows: on the $n$th of $\omega$-many rounds, player $\mathsf{Duplicator}$ plays a pair of natural numbers $i_n, j_n$, player $\mathsf{Spoiler}$ plays a pair of natural numbers $a_{2n}$, $b_{2n}$, and $\mathsf{Duplicator}$ plays a pair of natural numbers $a_{2n+1}$, $b_{2n+1}$.
$\mathsf{Duplicator}$ wins the length-$\omega$ play iff all of the following is true for each $n\in\omega$ (letting $(\mathcal{S}_n)_{n\in\mathsf{Tot}}$ be an appropriate list of the computable structures):
There are (not necessarily computable!) isomorphisms $$f: \mathcal{S}_{i_{n-1}}\rightarrow\mathcal{S}_{i_n},\quad g:\mathcal{S}_{j_{n-1}}\rightarrow\mathcal{S}_{j_n}$$ such that $f(a_k)=a_k$ and $g(b_k)=b_k$ for all $k<2n$. (To handle $n=0$ we'll assume $\mathcal{S}_{i_{-1}}=\mathcal{A}$ and $\mathcal{S}_{j_{-1}}=\mathcal{B}$.)
The substructures of $\mathcal{S}_{i_{n+1}}$ and $\mathcal{S}_{j_{n+1}}$ with domains $\{a_k: k<2n\}$ and $\{b_k: k<2n\}$ respectively are isomorphic.
Basically, this is a normal back-and-forth game of length $\omega$, but $\mathsf{Duplicator}$ is allowed to keep "scrambling" the structures involved as long as they don't mess with the already-labelled parts. When we require that all strategies involved be computable, this seems to complicate things:
Question: Suppose $\mathcal{A}$ and $\mathcal{B}$ are not computably isomorphic. Can $\mathsf{Duplicator}$ have a computable strategy $\Sigma$ which beats all computable strategies for $\mathsf{Spoiler}$ in the scrambled EF-game?
(Having $\mathsf{Spoiler}$ scramble things instead - or letting both players scramble! - would also be potentially interesting, I'm just focusing on this particular version to start with.)
