Rapid eye movements (REMs) are a defining characteristic of REM sleep during which vivid dreams occur. It has been suggested that REMs may be binocularly coordinated and related to "watching" dream images. For the first time, binocular eye movements were recorded during natural REM sleep in monkeys to test the conjugate nature of the oculomotor system and the "scanning hypothesis" of REMs during sleep. During REM sleep, the lines of sight of the two eyes are frequently misaligned up to 30 degrees horizontally and/or vertically. Since the lines of sight usually don't intersect, there is no fixation point. Instead, each eye is aimed at a different part of the visual field during REM sleep. Furthermore, REMs are not usually conjugate, but are disjunctive or even monocular in horizontal or vertical directions. These data argue against the idea that REMs actually "track" dream images, unless each eye is watching its own dream! Binocular misalignment and disjunctive (even monocular) REMs during sleep suggest that separate left eye and right eye pathways generate saccades in each eye and control the position of each eye. Binocular coordination cannot be the passive result of anatomical connectivity as has been argued previously, but instead must result from a high-level process associated with the awake state that coordinates activity in left-eye and right-eye pathways. Hering's law of equal innervation is not consistent with these data.