  We prove existence and uniqueness of strong solutions for a class of
semilinear stochastic evolution equations driven by general Hilbert
space-valued semimartingales, with drift equal to the sum of a linear maximal
monotone operator in variational form and of the superposition operator
associated to a random time-dependent monotone function defined on the whole
real line. Such a function is only assumed to satisfy a very mild symmetry-like
condition, but its rate of growth towards infinity can be arbitrary. Moreover,
the noise is of multiplicative type and can be path-dependent. The solution is
obtained via a priori estimates on solutions to regularized equations,
interpreted both as stochastic equations as well as deterministic equations
with random coefficients, and ensuing compactness properties. A key role is
played by an infinite-dimensional Doob-type inequality due to M\'etivier and
Pellaumail.
