Natural $\Pi^1_2$ (or worse) classes of structures? (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 $\Pi^1_1$? 
In particular:

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.)
 A: An example directly generalizing well-orderings is that the better quasi-orderings are $\Pi^1_2$-complete, shown in Marcone's aptly named The Set of Better Quasi-Orders is $\Pi^1_2$.  
A: An example that comes to mind is the set of recursive dilators; this is a $\Pi^1_2$-complete set (Theorem 4.1 in Girard 1985, Introduction to $\Pi^1_2$-logic). Dilators have some use outside of logic (e.g.: Some uses of dilators in combinatorial problems I, II, III).
Added October 27:
Definition: A dilator is functor from ON to ON (the large poset of ordinals and strictly increasing functions) preserving filtered colimits and pullbacks. A dilator is weakly finite if it maps finite ordinals to finite ordinals. A dilator is recursive if it is weakly finite and the morphism part is a recursive function from N to N under a standard coding of finite sequences as numbers.
Thus, weakly finite dilators are coded by reals, while recursive dilators are coded by numbers, and in the latter case forming a $\Pi^1_2$-complete set, just as the usual set of codes of recursive ordinals forms a $\Pi^1_1$-complete set.
