(To clarify, my interest is mainly lightface, that is, $\Pi^1_2$ instead of $\bf \Pi^1_2$, although it doesn't particularly matter.)
This is just an idle curiosity. In logic, I find myself frequently working with the countable ordinals; this is a $\Pi^1_1$ set of structures, in the sense that the set of reals coding well-orderings with domain $X\subseteq\omega$ is $\Pi^1_1$. This is also a fairly natural class of structures, of interest outside logic as well; and other interesting classes of structures - e.g., torsion groups, Artinian rings, infinite graphs into which a given infinite graph $G$ does not embed - are also $\Pi^1_1$ (and usually are vastly simpler than $\Pi^1_1$).
Obviously, there are more complicated classes of structures; however, all the examples I know are pretty artificial. So my question is: what are some natural classes of countable structures, which are worse than $\Pi^1_1$?
Is there a natural class of countable structures which is $\Pi^1_2$ complete?
(Of course, what "natural" means is subjective. What I mean by natural here is "of roughly equal interest to non-logicians as well-orderings," but examples from within logic would be okay if there is an interesting argument to be made that they will someday be of interest outside logic.)