Hadronic τ decays present an opportunity to determine the isovector part of the hadronic-vacuum-polarization contribution to the anomalous magnetic moment of the muon in a way complementary to e+e− → hadrons cross sections. However, the required isospin rotation is only exact in the isospin limit, and corrections need to be under control to draw robust conclusions, most notably for τ → ππντ decays to determine the two-pion contribution, \( {a}_{\mu}^{\textrm{HVP},\textrm{LO}}\left[\pi \pi, \tau \right] \) . In this work, we present a novel analysis of the required radiative corrections using dispersion relations, thereby extending in a model-independent way the previous analysis in chiral perturbation theory (ChPT) beyond the threshold region. In particular, we include the dominant structure-dependent virtual corrections from pion-pole diagrams, leading to sizable changes in the vicinity of the ρ(770) resonance. Moreover, we work out the matching to ChPT and devise a strategy for a stable numerical evaluation of real-emission contributions near the two-pion threshold, which proves important to capture isospin-breaking corrections enhanced by the threshold singularity. For the numerical analysis, we use a dispersive representation of the pion form factor including the ρ′, ρ′′ resonances, perform fits to the available data sets for the τ → ππντ spectral function, and calculate the corresponding radiative correction factor GEM(s) in a self-consistent manner. Based on these results, we evaluate the τ-specific isospin-breaking corrections to \( {a}_{\mu}^{\textrm{HVP},\textrm{LO}}\left[\pi \pi, \tau \right] \) .
