In PhysRevD.89.015021, it was shown that unknown masses in cascade decays can be measured more precisely using the full dimensionality of phase space instead of using one dimensional projections. Continuing this work, we consider the impact of a realistic collider environment on the previously proposed analysis. The effects we take into consideration include finite energy resolution, combinatorics that arise when the final state includes identical particles, and the presence of background events. We show that the multidimensional phase space analysis continues to yield improvements over measurements based on one-dimensional distributions in the presence of these effects. To deal with the presence of background, we use the novel technique of Voronoi tessellations that divides the multidimensional phase space into polygonal cells in order to implement analysis cuts that can recover the boundary of the kinematically allowed signal region and thereby allow for a clean signal-background separation.