  We consider obstacle problems for nonlinear stochastic evolution equations.
More precisely, the leading operator in our equation is a nonlinear, second
order pseudomonotone operator of Leray-Lions type. The multiplicative noise
term is given by a stochastic integral with respect to a Q-Wiener process. We
show well-posedness of the associated initial value problem for random initial
data on a bounded domain with a homogeneous Dirichlet boundary condition.
Therefore we consider a singular perturbation of our problem by a higher order
operator. Through the a priori estimates for the approximate solutions of the
singular perturbation, only weak convergence is obtained. This convergence is
not compatible with the nonlinearities in the equation. Therefore we use the
theorems of Prokhorov and Skorokhod to establish existence of martingale
solutions. Then, path-wise uniqueness follows from a L1-contraction principle
and we may apply the method of Gy\"ongy-Krylov to obtain stochastically strong
solutions. These well-posedness results serve as a basis for the study of
variational inequalities and Lewy-Stampacchia's inequalities for our problem.
