Here is a theorem of Hjorth, Kechris and Louveau which might be of some interest.

**Theorem**: Let $E^X_G$ be the orbit equivalence relation induced by a Borel action of a closed subgroup $G$ of $Sym(\mathbb{N})$ on some standard Borel space $X$. Then the following are equivalent:

i. $E^X_G$ is essentially countable.

ii. For some Polish topology $\tau$ on $X$ giving its Borel structure, $E^X_G$ is $\mathbf{\Sigma^0_2}$ in $(X,\tau)^2$.

iii. For some Polish topology $\tau$ on $X$ giving its Borel structure, $E^X_G$ is $\mathbf{\Sigma^0_3}$ in $(X,\tau)^2$.

(This is Theorem 1.12 in Jackson, Kechris and Louveau's paper "Countable Borel equivalence relations" and the following papers are given as references for this theorem: [Borel equivalence relations induced by actions of the symmetric group][1], [Borel equivalence relations and classifications of countable models][2]. Or, you can check out Chapter 12.5 of Gao's book *Invariant Descriptive Set Theory*.)

This theorem provides plenty of examples since the isomorphism relation on the standard Borel space of countable $L$-structures is given by the logic action of $Sym(\mathbb{N})$, where $L$ is a countable relational language (see Chapter 3.6 of Gao's book). For example, the isomorphism relation on finitely generated groups is essentially countable.

Of course, even though you might consider these examples "natural", you should keep in mind that we are actually changing the topology without changing the Borel structure, so the topologies which make these equivalence relations $\mathbf{\Sigma^0_3}$ might not be "natural".


  [1]: http://www.sciencedirect.com/science/article/pii/S0168007297000493
  [2]: http://www.sciencedirect.com/science/article/pii/S0168007296000061#