  In this paper we perform an amplitude analysis of $\eta'\to\pi^+\pi^-\gamma$
and confront it with the latest BESIII data. Based on the final-state
interaction theorem, we represent the amplitude in terms of an Omn\'es function
multiplied by a form factor that corresponds to the contributions from
left-hand cuts and right-hand cuts in the inelastic channels. We also take into
account the isospin violation effect induced by $\rho-\omega$ mixing. Our
results show that the anomaly contribution is mandatory in order to explain the
data. Its contribution to the decay width of $\Gamma(\eta'\to\pi\pi\gamma)$ is
larger than that induced by isospin violation. Finally we extract the pole
positions of the $\rho$ and $\omega$ as well as their corresponding residues.
