  We construct a martingale solution of the stochastic nonlinear Schr\"odinger
equation with a multiplicative noise of jump type in the Marcus canonical form.
The problem is formulated in a general framework that covers the subcritical
focusing and defocusing stochastic NLS in $H^1$ on compact manifolds and on
bounded domains with various boundary conditions. The proof is based on a
variant of the Faedo-Galerkin method. In the formulation of the approximated
equations, finite dimensional operators derived from the Littlewood-Paley
decomposition complement the classical orthogonal projections to guarantee
uniform estimates. Further ingredients of the construction are tightness
criteria in certain spaces of cadlag functions and Jakubowski's generalization
of the Skorohod-Theorem to nonmetric spaces.
