My favorite proof of the Cantor-Bernstein theorem is the one that argues by "histories" - given injections $f:A\rightarrow B$ and $g:B\rightarrow A$, we identify each element of $A$ as living in a maximal chain of type $\omega$, $\omega^*$, $\zeta$, or $n$ for some finite (even) $n$ and then reason by cases. This argument proves more than just the theorem itself: it shows that a bijection between $A$ and $B$ can be found within $f\cup g^{-1}$.
Now say that a structure $\mathfrak{A}$ (in the sense of universal algebra) is zigzag-intermediate iff
for every isomorphic disjoint $\mathfrak{B}$ and every pair of embeddings $f:\mathfrak{A}\rightarrow\mathfrak{B}, g:\mathfrak{B}\rightarrow\mathfrak{A}$ there is an isomorphism $h:\mathfrak{A}\cong\mathfrak{B}$ contained (set-theoretically) in the transitive symmetric closure of $f\cup g^{-1}$ as a relation on $\mathfrak{A}\sqcup\mathfrak{B}$, but
it is not the case that such an $h$ can always be found as a subset of $f\cup g^{-1}$.
My question is: do zigzag-intermediate structures exist?
This answer of Pace Nielsen suggests that structures which have any amount of "zigzag-ness" are very close to being sets (to my current reading, anyways); this question is an attempt to test this. My current instinct is that the answer should be no, and a positive answer would change how I interpret Pace's answer.
