We present a scheme for multiscale phase inversion (MPI) of seismic data that is less sensitive to the unmodeled physics of wave propagation and a poor starting model than standard full waveform inversion (FWI). To avoid cycle-skipping, the multiscale strategy temporally integrates the traces several times, i.e. high-order integration, to produce low-boost seismograms that are used as input data for the initial iterations of MPI. As the iterations proceed, higher frequencies in the data are boosted by using integrated traces of lower order as the input data. The input data are also filtered into different narrow frequency bands for the MPI implementation. At low frequencies, we show that MPI with windowed reflections approximates wave equation inversion of the reflection traveltimes, except no traveltime picking is needed. Numerical results with synthetic acoustic data show that MPI is more robust than conventional multiscale FWI when the initial model is far from the true model. Results from synthetic viscoacoustic and elastic data show that MPI is less sensitive than FWI to some of the unmodeled physics. Inversion of marine data shows that MPI is more robust and produces modestly more accurate results than FWI for this data set.