It has been suggested that to resolve ambiguities implicit in binocular perception of complex visual scenes, the brain adopts a continuity constraint assuming that disparities change smoothly with eccentricity. Stereoscopic transparency is characterized by abrupt changes of binocular disparity across retinal locations. The focus of the present study is how the brain uses the continuity constraint in the perception of stereoscopic transparency despite the presence of abrupt disparity changes. Observers viewed random-dot stereograms of overlapping transparent plane and cylindrical surfaces and had to distinguish between two orientations of the cylindrical surface under conditions of strictly controlled depth fixation. Surprisingly, maximal dot density of the transparent plane at which perception is still veridical dramatically decreases as depth separation between the surfaces grows. Persistence of this relationship, when binocular matching processes at each surface are separated to on and off brightness channels, suggests at least two stages in the underlying computation binocular matching and inter-surface interactions. We show that these phenomena cannot be accounted for by either higher severity of matching with high dot densities or the ability of the denser surface to pull vergence to its depth. We also measure contrast sensitivity and near-far symmetry of the underlying mechanism and propose a model of competitive interactions between dissimilar disparities.