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

Global well-posedness of the 4-d energy-critical stochastic nonlinear Schr\"{o}dinger equations with non-vanishing boundary condition