When two retinally adjacent image regions both claim ‘ownership’ of their common boundary based on different visual cues, their perceptual competition could result in: (1) cue averaging, in which the common boundary is not strongly perceived as owned by either region, or (2) perceptual bistability, in which the competing interpretations alternate in conscious perception over time. We report that when the perception of one or another illusory surface depends on the outcome of such a competition, the alternative percepts primarily exhibit bistability rather than averaging (or mutual weakening). More generally, we suggest that mutually inconsistent perceptual interpretations of sensory data will tend to exhibit bistability to the extent that they require significant constructive activity by vision. When one interpretation is more ‘literal’ (i.e. less constructive), it will tend to block alternative percepts. Put somewhat differently, when competing visual cues specify different preferred (but not necessary) interpretations, then the likely perceptual outcome is bistability rather than cue averaging. However, inconsistent visual cues can also result in perceptual bistability if the interpretations they specify are so incommensurable that simply averaging them would not provide useful information for perception.