Perceptual decisions are thought to depend on the activation of task-relevant neurons, whose activity is often correlated in time. Here, we examined how the temporal structure of shared variability in neuronal firing relates to perceptual choices. We recorded stimulus-selective neurons from visual area V5/MT while two monkeys (Macaca mulatta) made perceptual decisions about the rotation direction of structure-from-motion cylinders. Interneuronal correlations for a perceptually ambiguous cylinder stimulus were significantly higher than those for unambiguous cylinders or for random 2D motion during passive viewing. Much of the difference arose from correlations at relatively long timescales (hundreds of milliseconds). Choice-related neural activity (quantified as choice probability; CP) for ambiguous cylinders was positively correlated with interneuronal correlations and was specifically associated with their long timescale component. Furthermore, the slope of the long timescale - but not the instantaneous - component of the correlation predicted higher CPs towards the end of the trial i.e. close to the decision. Our results suggest that the perceptual stability of structure-from-motion cylinders may be controlled by enhanced interneuronal correlations on longer timescales. We propose this as a potential signature of top-down influences onto V5/MT processing that shape and stabilize the appearance of 3D-motion percepts.