<p>In this paper, we propose a splitting method for solving the two dimensional stochastic nonlinear Schrödinger equation with a damping term. Based on its Hamiltonian structure, the equation is split into two subproblems according to Strang splitting, with one of them being linear. The solution of the nonlinear subproblem is computed exactly, while the linear subproblem is discretized using the Fourier pseudo-spectral method in the spatial direction and the implicit midpoint method with respect to the time direction. Through theoretical analysis and proofs, we demonstrate the superior performance of this fully discrete scheme in preserving the system’s conservation laws and ensuring convergence. Numerical experiments on single and double solitons under varying damping confirm first-order time accuracy and spectral precision in space as predicted by the theory.</p>