The beam-helicity asymmetry was measured, for the first time, in photoproduction of $\pi^{0}\eta$ pairs on carbon, aluminum, and lead, with the A2 experimental setup at MAMI. The results are compared to an earlier measurement on a free proton and to the corresponding theoretical calculations. The Mainz model is used to predict the beam-helicity asymmetry for the nuclear targets. The present results indicate that the photoproduction mechanism for $\pi^{0}\eta$ pairs on nuclei is similar to photoproduction on a free nucleon. This process is dominated by the $D_{33}$ partial wave with the $\eta\Delta(1232)$ intermediate state.