Determinantal point processes (DPPs for short) are a class of repulsive point processes. They have found some statistical applications to model spatial point pattern datasets with repulsion between close points. In the case of DPPs on finite sets, they are defined by a matrix called the DPP kernel which is usually assumed to be symmetric. While there are a few known examples of DPPs with nonsymmetric kernels, not much is known on how this affects their usual properties. In this paper, we demonstrate how to adapt the results on $P_0$ matrices to the DPP setting in order to get necessary and sufficient conditions for the well-definedness of DPPs with nonsymmetric kernels. We also generalize various common results on DPPs. We then show how to use these results to construct attractive couplings of regular DPPs with symmetric kernels in order to model spatial marked point patterns with repulsion between points of the same mark and attraction between points of different marks.