We investigate the phase-space dynamics of a hydrogenic atom in a very intense, high-frequency laser field, by comparing the time-dependent quantum Wigner functions with the corresponding classical distributions calculated using the Monte Carlo method. In both cases we model the atom by a one-dimensional soft-core potential in order to simplify the calculation. We demonstrate the importance of nonclassical interferences in the Wigner function and show how negative parts of it observed at the peak of the pulse are associated with the formation of a coherent superposition of dressed ‘‘Kramers-Henneberger’’ (KH) eigenstates. These signatures of the quantum dynamics are eliminated towards the end of the pulse when the coherence between the KH eigenstates degrades so that the classical and the quantum mechanical phase-space distributions come into agreement.