  We prove pathwise uniqueness for solutions of the nonlinear Schr\"{o}dinger
equation with conservative multiplicative noise on compact 3D manifolds. In
particular, we generalize the result by Burq, G\'erard and Tzvetkov (N. Burq,
P. G\'erard, and N. Tzvetkov. Strichartz inequalities and the nonlinear
Schr\"{o}dinger equation on compact manifolds. American Journal of Mathematics,
126 (3):569--605, 2004) to the stochastic setting. The proof is based on
deterministic and stochastic Strichartz estimates and the Littlewood-Paley
decomposition.
