We formulate computationally efficient classical stochastic measurement trajectories for a multimode quantumsystem under continuous observation. Specifically, we consider the nonlinear dynamics of an atomic Bose-Einstein condensate contained within an optical cavity subject to continuous monitoring of the light leakingout of the cavity. The classical trajectories encode within a classical phase-space representation a continuousquantum measurement process conditioned on a given detection record. We derive a Fokker-Planck equationfor the quasiprobability distribution of the combined condensate-cavity system. We unravel the dynamics intostochastic classical trajectories that are conditioned on the quantum measurement process of the continuouslymonitored system. Since the dynamics of a continuously measured observable in a many-atom system can beclosely approximated by classical dynamics, the method provides a numerically efficient and accurate approachto calculate the measurement record of a large multimode quantum system. Numerical simulations of thecontinuously monitored dynamics of a large atom cloud reveal considerably fluctuating phase profiles betweendifferent measurement trajectories, while ensemble averages exhibit local spatially varying phase decoherence.Individualmeasurement trajectories lead to spatial pattern formation and optomechanical motion that solely resultfrom the measurement backaction. The backaction of the continuous quantum measurement process, conditionedon the detection record of the photons, spontaneously breaks the symmetry of the spatial profile of the condensateand can be tailored to selectively excite collective modes.