Actions of tame abelian product groups

A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. Suppose that G=\prod_n \Gamma_n is a product of countable abelian groups. It follows from results of Solecki and Ding-Gao that if G is tame, then all of its actions are in fact potentially \Pi^0_6. Ding and Gao conjectured that this bound could be improved to \Pi^0_3. We refute this, by finding an action of a tame abelian product group, which is not potentially \Pi^0_5.
The proof involves forcing over models where the axiom of choice fails for sequences of finite sets.
This is joint work with Shaun Allison.