When the two eyes’ processing streams meet in visual cortex, two things can happen: sufficiently similar monocular inputs are combined into a fused representation, whereas markedly different inputs engage in rivalry. Interestingly, the emergence of rivalry appears to require attention. Withdrawing attention causes the alternating monocular dominance that characterizes rivalry to cease, apparently allowing both monocular signals to be processed simultaneously. What happens to these signals in this case, however, remains something of a mystery; are they fused into an integrated representation? In a set of experiments, we show this not to be the case: visual aftereffects are consistent with the simultaneous yet separate presence of two segregated monocular representations, rather than a joint representation. These results provide evidence that dichoptic vision without attention prompts a third and previously unknown mode, where both eyes’ inputs receive equal processing, but escape interocular fusion.