We adopt a novel formalism for the low-energy analysis of the γp → K + πΣ photoproduction reaction to calculate the πΣ invariant mass distributions in the Λ(1405) resonance region. The approach adheres to constraints arising from unitarity, gauge invariance and chiral perturbation theory, and is used without adjusting any model parameters. It is found that the meson-baryon rescattering in the final state has a major impact on the magnitude and structure of the generated spectra that are compared with the experimental data from the CLAS collaboration. We demonstrate a large sensitivity of the theoretical predictions to the choice of the coupled-channel πΣ − KN model amplitudes which should enable one to constrain the parameter space of these models.