  We consider the energy-critical stochastic cubic nonlinear Schr\"odinger
equation on $\mathbb R^4$ with additive noise, and with the non-vanishing
boundary conditions at spatial infinity. By viewing this equation as a
perturbation to the energy-critical cubic nonlinear Schr\"odinger equation on
$\mathbb R^4$, we prove global well-posedness in the energy space. Moreover, we
establish unconditional uniqueness of solutions in the energy space.
