  We establish well-posedness in the mild sense for a class of stochastic
semilinear evolution equations on $L_p$ spaces, driven by multiplicative Wiener
noise, with a drift term given by an evaluation operator that is assumed to be
quasi-monotone and polynomially growing, but not necessarily continuous. In
particular, we consider a notion of mild solution ensuring that the evaluation
operator applied to the solution is still function-valued, but satisfies only
minimal integrability conditions. The proofs rely on stochastic calculus in
Banach spaces, monotonicity and convexity techniques, and weak compactness in
$L_1$ spaces.
