In tropical geometry, a tropical hyperplane in (d-1)-space has d oriented sides called 'sectors'. Analogous to the covectors of realizable oriented matroids, an arrangement of labeled tropical hyperplanes partitions (tropical) space into cells determined by the sectors that a point lies in relative to the each hyperplane. These 'type' decompositions were first studied by Develin and Sturmfels, where in particular a close connection to regular triangulations of products of simplices was established. In joint work with Michael Joswig and Raman Sanyal we consider a coarsening of this data in which the labels of the hyperplanes are neglected. We study the combinatorics and geometry of the resulting coarse type decompositions and use this to obtain cellular resolutions of certain monomial ideals associated to the arrangement (originally this was the motivation for studying coarse types). We discuss how these coarse type ideals (and their Alexander duals) are related to other well-studied objects.