We present high-transverse field measurements, as a function of temperature, in monoclinic ZrO2 and HfO2. In monoclinic zirconia and hafnia, a diamagnetic component had been previously reported in low-transverse-fields, but a significant fraction of the total muon polarization was missing in these experiments. We now characterize this missing fraction using the high-field capabilities at TRIUMF: a high relaxation component (above 100 μs-1 in monoclinic ZrO2, about 10 μs-1 in HfO2) is observed, which we relate to the formation of compact muonium in these materials. A model for the formation of muonium in these materials is presented.