Most finite spatial point process models specified by a density are locally stable, implying that the Papangelou intensity is bounded by some integrable function $\beta$ defined on the space for the points of the process. It is possible to superpose a locally stable spatial point process $X$ with a complementary spatial point process $Y$ to obtain a Poisson process $X \cup Y$ with intensity function $\beta$. Underlying this is a bivariate spatial birth-death process $(X_t, Y_t)$ which converges towards the distribution of $(X, Y)$. We study the joint distribution of $X$ and $Y$, and their marginal and conditional distributions. In particular, we introduce a fast and easy simulation procedure for $Y$ conditional on $X$. This may be used for model checking: given a model for the Papangelou intensity of the original spatial point process, this model is used to generate the complementary process, and the resulting superposition is a Poisson process with intensity function $\beta$ if and only if the true Papangelou intensity is used. Whether the superposition is actually such a Poisson process can easily be examined using well known results and fast simulation procedures for Poisson processes. We illustrate this approach to model checking in the case of a Strauss process.