We develop a classical theoretical description for nonlinear many-body dynamics that incorporates the back-action of a continuous measurement process. The classical approach is compared with the exact quantum solution in an example with an atomic Bose–Einstein condensate in a double-well potential where the atom numbers in both potential wells are monitored by light scattering. In the classical description, the back-action of the measurements appears as diffusion of the relative phase of the condensates on each side of the trap. When the measurements are frequent enough to resolve the system dynamics, the system behaves classically. This happens even deep in the quantum regime, and demonstrates how classical physics emerges from quantum mechanics as a result of measurement back-action