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 ρ(770) and ω(782) resonances. This particular IB contribution to the pion vector form factor and thus the line shape in e⁺e⁻→ π⁺π⁻ can be described by the residue at the ω pole — the ρ-ω mixing parameter ϵ_ω. Here, we argue that while in general analyticity requires this parameter to be real, the radiative channels π⁰γ, ππγ, ηγ can induce a small phase, whose size we estimate as δ_ϵ = 3.5(1.0)° by using a narrow-width approximation for the intermediate-state vector mesons. We then perform fits to the e⁺e⁻→ π⁺π⁻ 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 ρ-ω mixing, and the mass of the ω resonance. We find that the global fit does prefer a non-vanishing value of δ_ϵ = 4.5(1.2)°, close to the narrow-resonance expectation, but with a large spread among the data sets, indicating systematic differences in the ρ-ω region.