  We prove global well-posedness for a class of dissipative semilinear
stochastic evolution equations with singular drift and multiplicative Wiener
noise. In particular, the nonlinear term in the drift is the superposition
operator associated to a maximal monotone graph everywhere defined on the real
line, on which no continuity nor growth assumptions are imposed. The hypotheses
on the diffusion coefficient are also very general, in the sense that the noise
does not need to take values in spaces of continuous, or bounded, functions in
space and time. Our approach combines variational techniques with a priori
estimates, both pathwise and in expectation, on solutions to regularized
equations.
