I have been reading about Borel equivalence relations and I have noticed that while $\bf\Sigma^0_3$ equivalence relations are mentioned, there is a conspicuous absence of natural examples (other than $F_{\sigma}$ or $G_{\delta}$ ones).

For example, in Gao's *Invariant Descriptive Set Theory* he provides examples of $\bf\Pi^0_3$ equivalence relations and even proves that one, $E_0^{\omega}$, is not reducible to any $\bf\Sigma_3^0$ equivalence relation. But he provides no examples of these.

In Kanovei's *Borel Equivalence Relations*, he proves that all equivalence relations with $\bf\Sigma^0_3$ classes are pinned (exactly what this means isn't important). But again, there are no examples mentioned except those of lower complexity.

Of course examples can be constructed. If $X$ is Polish and $A\subset X$ is $\bf\Sigma_3^0$, you can look at the equivalence relation on $X\times 2$ relating vertical sections with first coordinate in $A$. But this has a Borel selector, so it is smooth.

So, are there natural examples of $\bf\Sigma^0_3$ equivalence relations? Or do all such equivalence relations have to be simple, like how all $G_{\delta}$ equivalence relations are smooth? Or something else? I would be curious to hear about anything that can be said concerning these ERs.