Current notions of binocular depth perception include (1) neural computations that solve the correspondence problem and calculate retinal positional disparity, and (2) recovery of ecologically valid occlusion relationships. The former framework works well for stimuli with unambiguous interocular correspondence, but less so for stimuli without well-defined disparity cues. The latter framework has been proposed to account for the result of perceived depth in stimuli without interocular correspondence, but its mechanism remains unclear. In order to obtain more insight into the mechanism, we studied the depth percept elicited by a family of stereograms -- "sieve" stimuli, adapted from Howard (Perception 24, 67-74 (1995)) -- with interocular differences but no well-defined positional disparity cue. The perceived depth was measured by comparison to references at various depths established by standard retinal disparity and was consistently found to lie behind the fixation plane. Moreover, the magnitude of the depth percept depended on both the horizontal and vertical spatial characteristics of the stimulus in ways that were at odds with constraints of occlusion geometry. In comparison to the depth percept elicited by stimuli with well-defined disparity cues, the precision of the percept from the sieve stimuli was 10 to 20 times worse, suggesting that a different underlying computation was involved. Thus, neither of the above frameworks accounts for the depth percept arising from these stimuli. We discuss implications of our results for physiologically based computations underlying binocular depth perception.