  Isospin-breaking (IB) effects in the two-pion contribution to hadronic vacuum
polarization (HVP) can be resonantly enhanced, if related to the interference
of the $\rho(770)$ and $\omega(782)$ resonances. This particular IB
contribution to the pion vector form factor and thus the line shape in
$e^+e^-\to \pi^+\pi^-$ can be described by the residue at the $\omega$ pole -
the $\rho$-$\omega$ mixing parameter $\epsilon_\omega$. Here, we argue that
while in general analyticity requires this parameter to be real, the radiative
channels $\pi^0\gamma$, $\pi\pi\gamma$, $\eta\gamma$ can induce a small phase,
whose size we estimate as $\delta_\epsilon=3.5(1.0)^\circ$ by using a
narrow-width approximation for the intermediate-state vector mesons. We then
perform fits to the $e^+e^-\to \pi^+\pi^-$ data base and study the consequences
for the two-pion HVP contribution to the anomalous magnetic moment of the muon,
its IB part due to $\rho$-$\omega$ mixing, and the mass of the $\omega$
resonance. We find that the global fit does prefer a non-vanishing value of
$\delta_\epsilon=4.5(1.2)^\circ$, close to the narrow-resonance expectation,
but with a large spread among the data sets, indicating systematic differences
in the $\rho$-$\omega$ region.
