This question is a kind of dual to this earlier one. Note that if we replace $\mathsf{FOL}$ with $\mathcal{L}_{\omega_1,\omega}$, things trivialize since we can use the theory $\{\varphi^A\leftrightarrow\varphi^B: \varphi\in\mathcal{L}_{\omega_1,\omega}\}$. The situation would also trivialize if we replaced $\mathsf{FOL}$ with $\mathsf{SOL}$, but not trivially: the point is that the game characterization by Vaananen and Wang is appropriately second-order expressible. Throughout, all signatures are finite.
Fix two unary relation symbols $A,B$ and a binary relation symbol $E$ (I'll call $\{E\}$-structures "graphs" for simplicity). For every first-order theory $T$ in a language $\supseteq\{A,B,E\}$, let $\sim_T$ be the relation on graphs given by $$\mathfrak{A}\sim_T\mathfrak{B}\quad\iff\quad\exists\mathfrak{M}\models T(A^\mathfrak{M}\upharpoonright\{E\}\cong\mathfrak{A}\wedge B^\mathfrak{M}\upharpoonright\{E\}\cong\mathfrak{B}).$$
Letting $\sim$ be the intersection of all $\sim_T$s which are equivalence relations coarsening $\equiv_{\omega_1,\omega}$, we get that $\sim$ is an equivalence relation on graphs which is isomorphism-invariant and (by a result of Farmer S.) strictly coarsens $\equiv_{\omega_1,\omega}$ on graphs in general.
Question: Is $\sim$ strictly coarser than $\equiv_{\omega_1,\omega}$ (or equivalently, $\cong$) on countable graphs?
That is, are there non-isomorphic countable graphs which are $\sim$-equivalent? Note that the obvious theory describing an isomorphism (or even potential isomorphism) between the $A$- and $B$-parts does not satisfy the $\equiv_{\omega_1,\omega}$-coarsening property.
