We study the defocusing energy critical nonlinear wave equation in four dimensions. Our main result proves the stability of the scattering mechanism under random perturbations of the initial data. The random perturbation is defined through a microlocal randomization, which is based on a unit-scale decomposition in physical and frequency space. In contrast to the previous literature, we do not require the spherical symmetry of the perturbation.

The main novelty lies in a wave packet decomposition of the random linear evolution. Through this decomposition, we can adaptively estimate the interaction between the rough and regular components of the solution. Our argument relies on techniques from restriction theory, such as Bourgain's bush argument and Wolff's induction on scales.