Every Borel-overt fiberwise quasi-Polish groupoid is Borel equivalent to an open Polish groupoid, hence to an action groupoid of a Polish group action; Borel-overt classwise Polish equivalence relations are Borel bireducible with orbit equivalence relations of free Polish group actions.
Notes on quasi-Polish spaces
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abstract
Quasi-Polish spaces were introduced by de Brecht as a possibly non-Hausdorff generalization of Polish spaces sharing many of their descriptive set-theoretic properties. We give a self-contained exposition of the basic theory of quasi-Polish spaces, based on their "logical" characterization as $\mathbf{\Pi}^0_2$ subspaces of countable powers of Sierpinski space, with several new proofs emphasizing this point of view as well as making more extensive use of Baire category techniques.
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Componentwise Polish groupoids and equivalence relations
Every Borel-overt fiberwise quasi-Polish groupoid is Borel equivalent to an open Polish groupoid, hence to an action groupoid of a Polish group action; Borel-overt classwise Polish equivalence relations are Borel bireducible with orbit equivalence relations of free Polish group actions.