Learning self-supervised visual correspondence is a long-studied task fundamental to visual understanding and human perception. However, existing correspondence methods largely focus on small image transformations, such as object tracking in high-framerate videos or learning pixel-to-pixel mappings between images with high view overlap. This severely limits their application in dynamic multi-view settings such as robot imitation learning or augmented reality. In this work, we introduce Predictive Cycle Consistency for learning object correspondence between extremely disjoint views of a scene without paired segmentation data. Our technique bootstraps object correspondence pseudolabels from raw image segmentations using conditional grayscale colorization and a cycle-consistency refinement prior. We then train deep ViTs on these pseudolabels, which we use to generate higher-quality pseudolabels and iteratively train better correspondence models. We demonstrate the performance of our method under both extreme in-the-wild camera view changes and across large temporal gaps in video. Our approach beats all prior supervised and prior SoTA self-supervised correspondence models on the EgoExo4D correspondence benchmark (+6.7 IoU Exo Query) and the prior SoTA self-supervised methods SiamMAE and DINO V1&V2 on the DAVIS-2017 and LVOS datasets across large frame gaps.