The second-order nonlinear response of bulk GaAs in the terahertz (THz) range is mapped via the THz field emitted after near-infrared interband excitation. Phase-resolved THz detection reveals three nonlinear processes occurring in parallel, the Raman excitation of transverse optical phonons, the creation of coherent polarizations on heavy-hole-light-hole transitions, and the generation of displacive shift currents with a THz spectrum controlled by the near-infrared optical phase. Theoretical calculations reproduce the data and demonstrate the interband character of shift currents.