Gratings with different disparities are sometimes seen as transparent surfaces, each with a distinct depth, when they are superimposed, and sometimes they are seen as a coherent plaid confined to a single depth plane—stereo analogs of transparent and coherent motion. Briefly presented sinusoidal gratings of similar spatial frequencies are seen to cohere in depth. The resulting plaid generally appears in a depth plane different from that of either component grating viewed separately; the plaid may even appear on the oppose side of fixation from the component gratings. Under similar viewing conditions, squarewave gratings are typically seen as transparent. Objective measures, gathered here using depth-order discriminations, show that the perception of transparency between squarewave gratings requires a minimum disparity difference that varies with the gratings’ orientations. Gratings that are near orthogonal in orientation, or that give the plaid a near-horizontal disparity, favor the perception of coherence. Gratings that form a plaid having a large ratio of vertical to horizontal disparities favor the perception of transparency. The data are consistent with a Bayesian prior favoring single surfaces when disparities are small and near-horizontal. Disparities that are large or non-horizontal are more likely to be aperture disparities that result from viewing separate but overlapping surfaces. The sinewave-squarewave difference leads to the conclusion that coherence between components is required both for seeing a broadband pattern in a single depth plane and for seeing it in a different depth plane from other superimposed patterns.