We describe the geometry coarse convex hulls in coarse median spaces. When the space comes equipped with a family of projection maps to hyperbolic spaces satisfying certain conditions, we show that coarse hulls are quasi-isometric to CAT(0) cube complexes. From this we deduce a distance formula, and show the existence of monotone paths connecting pairs of points. In particular, we recover a number of results about hierarchically hyperbolic spaces.
Preprint, Warwick, August 2018. Revised, October 2020.