Context. For several reasons, methanol is believed to be formed on grain surfaces and, in warm environments, released in the gas phase. In the past, multiply deuterated isotopologues of methanol have been detected in gas phase around several low-mass protostars. In all these sources, there is significantly more CH$_2$DOH than CH$_3$OD. Various hypotheses have been suggested to explain this anomaly, but none is fully convincing.Aims. In this work, we test a new hypothesis experimentally: the spontaneous exchange between hydrogen and deuterium atoms in water ice as responsible for the deficiency of CH$_3$OD with respect to CH$_2$DOH.Methods. We follow the temperature dependence of the composition of interstellar ice analogs initially composed of CD$_3$OD and H$_2$O. To this aim, thin films of intimate H$_2$O:CD$_3$OD ice mixtures, condensed at low temperature ($<$110 K), are monitored by Fourier transform infrared (FTIR) spectroscopy up to the complete evaporation of CD$_3$OD (~170 K).Results. Rapid hydrogen/deuterium (H/D) exchange is observed, at 120 K and above, through the growth of the $ν_{\rm OD}$ stretching mode of HDO at ~2425 cm$^{-1}$. It is also shown that H/D exchange occurs i) on the hydroxyl functional group of methanol, i.e through hydrogen bonds, and ii) before the completion of crystallization.Conclusions. The present results suggest that the much lower abundance of CH$_3$OD compared to CH$_2$DOH in low-mass protostars could reflect H/D exchanges in water ice either prior to or definitely during the grain mantle sublimation. This solid-state depletion mechanism, so far neglected in the astronomical literature, might affect other deuterated molecules with hydrogen bonds.