  We introduce a stochastic spin-wave theory tailored to describe quantum
trajectories in continuously monitored long-range interacting spin systems. Our
method, based on the bosonization of spin-wave excitations on top of a strong
collective polarization, enables the efficient simulation of large-scale
interacting spins, offering insights into nonlinear features of the dynamics
such as entanglement and trajectory correlations. We showcase the versatility
of our framework by exploring an entanglement phase transition in a monitored
spin system with power-law interactions and dwelling on how our method
mitigates the experimental challenges of post-selection in detecting monitored
quantum phases.
