  A Polish group $G$ is tame if for any continuous action of $G$, the
corresponding orbit equivalence relation is Borel. When $G = \prod_n \Gamma_n$
for countable abelian $\Gamma_n$, Solecki (1995) gave a characterization for
when $G$ is tame. Ding and Gao (2017) showed that for such $G$, the orbit
equivalence relation must in fact be potentially $\mathbf{\Pi}^0_6$, while
conjecturing that the optimal bound could be $\mathbf{\Pi}^0_3$. We show that
the optimal bound is $D(\mathbf{\Pi}^0_5)$ by constructing an action of such a
group $G$ which is not potentially $\mathbf{\Pi}^0_5$, and show how to modify
the analysis of Ding and Gao to get this slightly better upper bound. It
follows, using the results of Hjorth, Kechris, and Louvaeu (1998), that this is
the optimal bound for the potential complexity of actions of tame abelian
product groups. Our lower-bound analysis involves forcing over models of set
theory where choice fails for sequences of finite sets.
