Is there a back-and-forth condition equivalent to elimination of the $\exists^\infty$ quantifier? (The question may not have much of a sense. Then, please, argue why.)
Background. For instance, the back-and-forth condition for elimination of (classical) quantifiers reads as follows. For every partial isomorphism $f:M\to N$ and every $a\in M$ there is $N'\succeq N$ and partial isomorphism $g:M\to N'$ that extends $f$ and is defined on $a$.
A theory $T$ eliminates the $\exists^\infty$ quantifier if for every partitioned$^{*}$ formula $\psi(x,y)$ there is an $n$ such that for all $m>n$, $\ T\ \vdash\ \exists^{\ge m}y\;\psi(x,y)\leftrightarrow\exists^{\ge n}y\;\psi(x,y)$.
($^{*}$) By partitioned formula I mean a formula and a pair of tuples of variables.
