We investigate the real-time dynamics of U (1) and SU (N) gauge theories coupled to fermions on a lattice. While real-time lattice gauge theory is not amenable to standard importance sampling techniques, for a large class of time-dependent problems the quantum dynamics can be accurately mapped onto a classical-statistical ensemble. We illustrate the genuine quantum contributions included in this description by giving a diagrammatic representation in a series expansion. The nonperturbative simulation method is then applied to electron-positron production in quantum electrodynamics in three spatial dimensions. We compare to analytic results for constant background field and demonstrate the importance of backreaction of the produced fermion pairs on the gauge fields.