  We prove existence and uniqueness of strong solutions for a class of
second-order stochastic PDEs with multiplicative Wiener noise and drift of the
form $\operatorname{div} \gamma(\nabla \cdot)$, where $\gamma$ is a maximal
monotone graph in $\mathbb{R}^n \times \mathbb{R}^n$ obtained as the
subdifferential of a convex function satisfying very mild assumptions on its
behavior at infinity. The well-posedness result complements the corresponding
one in our recent work arXiv:1612.08260 where, under the additional assumption
that $\gamma$ is single-valued, a solution with better integrability and
regularity properties is constructed. The proof given here, however, is
self-contained.
