  We perform an effective field theory analysis of the $\tau^- \to \pi^- \pi^0
\nu_\tau$ decays, that includes the most general interactions between Standard
Model fields up to dimension six, assuming left-handed neutrinos. We constrain
as much as possible the necessary Standard Model hadronic input using chiral
symmetry, dispersion relations, data and asymptotic QCD properties. As a
result, we set precise (competitive with low-energy and LHC measurements)
bounds on (non-standard) charged current tensor interactions, finding a very
small preference for their presence, according to Belle data. Belle-II near
future measurements can thus be very useful in either confirming or further
restricting new physics tensor current contributions to these decays. For this,
the spectrum in the di-pion invariant mass turns out to be particularly
promising. Distributions in the angle defined by the $\tau^-$ and $\pi^-$
momenta can also be helpful if measured with less than $10\%$ accuracy, both
for non-standard scalar and tensor interactions.
