*This was asked and bountied at MSE without success.*

Throughout, we work in $\mathsf{ZF}$.

Say that a set $X$ is $\Pi^1_1$-pseudofinite if for every first-order sentence $\varphi$, if $\varphi$ has a model with underlying set $X$ then $\varphi$ has a finite model. *(See here, and the answer and comments, for background.)* Every $\Pi^1_1$-pseudofinite set is Dedekind-finite basically trivially, and with some model theory we can show that every amorphous set is $\Pi^1_1$-pseudofinite. Beyond that, however, things are less clear.

In particular, I noticed that I can't seem to prove a very basic property of this notion:

Is the union of two $\Pi^1_1$-pseudofinite sets always $\Pi^1_1$-pseudofinite?

I'm probably missing something simple, but I don't see a good way to get a handle on this. A structure on $X=A\sqcup B$ might not "see" that partition at all, and so none of the simple tricks I can think of work.