Now asked at MO.
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.
To move this off the unanswered queue, I'll mention here that Harry West gave a negative answer to this question at MO. (I've made this CW to avoid reputation gain.)