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Coherent photoproduction of the neutral pion and eta meson on the deuteron, $γ{d}${$\to$}$π^0η{d}$, has been experimentally studied at incident photon energies ranging from the reaction threshold to 1.15 GeV. The total cross section demonstrates a rapid rise below 1 GeV. The data are underestimated by the existing theoretical calculations based on quasi-free $π^0 η$ photoproduction on the nucleon followed by deuteron coalescence. At the same time, the data are rather well reproduced by the calculations taking into account the final-state interaction. We have also measured for the first time the differential cross sections: the $π^0 η$ invariant-mass distribution $dσ/dM_{πη}$, the $π^0 d$ invariant-mass distribution $dσ/dM_{πd}$, the $ηd$ invariant-mass distribution $dσ/dM_{ηd}$, and the distribution over the deuteron emission angle $dσ/dΩ_d$ in the overall center-of-mass frame. The measured cross section $dσ/dΩ_d$ does not exhibit strongly backward-peaking behavior predicted by the calculations. At all incident energies, an increase in $dσ/dM_{ηd}$ near the $ηd$ threshold is observed, which indicates a bound or virtual $ηd$ state resulting from a strong attraction between $η$ and a deuteron. The possibilities of using coherent $π^0η$ photoproduction on a nucleus to study the $η$-nuclear interaction are also discussed.