Many signal processing methods break the target signal into overlapping patches, process them separately, and then stitch them back to produce an output. How to merge the resulting patches at the overlaps is central to such methods. We propose a novel framework for this type of problem based on the idea that estimated patches should coincide at the overlaps (consensus), and develop an algorithm for solving the general problem. In particular, an efficient method for projecting patches onto the consensus constraint is presented. Finally, a simple and effective audio/image/video inpainting algorithm is developed using this framework.