Tertiary interactions between irregular incident waves and their reflections from a rectangular box are investigated experimentally. New experiments feature uni-directional seas with normal and oblique incidence to the side of the box and spread seas, where data is notably lacking in the literature. NewWave conditioning analysis is applied to show that the delays in maximum response associated with tertiary interactions are present in front of the box for both uni-directional and spread sea states. However, only in the uni-directional sea states with normal incidence is there significant amplification of free-surface elevation in front of the box beyond the linear diffraction levels. The nature of the large nonlinear run-up in this case is studied in detail. As an important finding, this work shows that localised tertiary interactions, whilst an interesting physical phenomenon, are unlikely to be important for wave–structure interactions in general, in realistic open ocean wave conditions.