  We study the $\tau^- \to \nu_{\tau} \pi^{-}\pi^{0}\ell^{+}\ell^{-}$
($\ell=e,\,\mu$) decays, which are $O(\alpha^2)$-suppressed with respect to the
dominant di-pion tau decay channel. Both the inner-bremsstrahlung and the
structure- (and model-)dependent contributions are considered. In the $\ell=e$
case, structure-dependent effects are $\mathcal{O}(1\%)$ in the decay rate,
yielding a clean prediction of its branching ratio, $2.3\times10^{-5}$,
measurable with BaBar or Belle(-II) data. For $\ell=\mu$, both contributions
have similar magnitude and we get a branching fraction of
$(1.6\pm0.3)\times10^{-7}$, reachable by the end of Belle-II operation. These
decays allow to study the dynamics of strong interactions with simultaneous
weak and electromagnetic probes; their knowledge will contribute to reducing
backgrounds in lepton flavor/number violating searches.
