  Puzzles in the determination of the hadronic-vacuum-polarization contribution
currently impede a conclusive interpretation of the precision measurement of
the anomalous magnetic moment of the muon at the Fermilab experiment. One such
puzzle concerns tensions between evaluations in lattice QCD and using
$e^+e^-\to\text{hadrons}$ cross-section data. In lattice QCD, the dominant
isospin-symmetric part and isospin-breaking (IB) corrections are calculated
separately, with very different systematic effects. Identifying these two
pieces in a data-driven approach provides an opportunity to compare them
individually and trace back the source of the discrepancy. Here, we estimate
the IB component of the lattice-QCD calculations from phenomenology, based on a
comprehensive study of exclusive contributions that can be enhanced via
infrared singularities, threshold effects, or hadronic resonances, including,
for the first time, in the $e^+e^-\to3\pi$ channel. We observe sizable
cancellations among different channels, with a sum that even suggests a
slightly larger result for the QED correction than obtained in lattice QCD. We
conclude that the tensions between lattice QCD and $e^+e^-$ data therefore
cannot be explained by the IB contributions in the lattice-QCD calculations.
