The beam-helicity asymmetry in associated electroproduction of real photons, ep → eγπN, in the ∆(1232)-resonance region is measured using the longitudinally polarized Hera positron beam and an unpolarized hydrogen target. Azimuthal Fourier amplitudes of this asymmetry are extracted separately for two channels, ep → eγπ 0 p and ep → eγπ + n, from a data set collected with a recoil detector. All asymmetry amplitudes are found to be consistent with zero.