  We consider the nonlinear Schr\"odinger equation with multiplicative spatial
white noise and an arbitrary polynomial nonlinearity on the two-dimensional
full space domain. We prove global well-posedness by using a gauge-transform
introduced by Hairer and Labb\'e (2015) and constructing the solution as a
limit of solutions to a family of approximating equations. This paper extends a
previous result by Debussche and Martin (2019) with a sub-quadratic
nonlinearity.
