An extension of the popular log-Gaussian Cox process (LGCP) model for spatial point patterns is proposed for data exhibiting fundamentally different behaviors in different subregions of the spatial domain. The aim of the analyst might be either to identify and classify these regions, to perform kriging, or to derive some properties of the parameters driving the random field in one or several of the subregions. The extension is based on replacing the latent Gaussian random field in the LGCP by a latent spatial mixture model specified using a categorically valued random field. This classification is defined through level set operations on a Gaussian random field and allows for standard stationary covariance structures, such as the Matérn family, to be used to model random fields with some degree of general smoothness but also occasional and structured sharp discontinuities. A computationally efficient MCMC method is proposed for Bayesian inference and we show consistency of finite dimensional approximations of the model. Finally, the model is fitted to point pattern data derived from a tropical rainforest on Barro Colorado island, Panama. We show that the proposed model is able to capture behavior for which inference based on the standard LGCP is biased.