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We consider the problem of recovering a complex vector (up to a global unimodular constant) given noisy and incomplete outer product measurements. Such problems arise when implementing distributed clock synchronization schemes, radar autofocus methods, and phaseless signal recovery. This problem is known as vector synchronization and is a variant of the more common angular synchronization problem. In applications with windowed measurements and/or convolutional models - for example, phase retrieval from STFT magnitude data, the outer product measurement matrix is highly incomplete and has a block diagonal structure. We describe a vector synchronization technique which applies an eigenvector computation to blocks of this matrix followed by a block compatibility operation to piece together the final solution. We provide theoretical guarantees (in the noiseless case) and empirical simulations demonstrating the accuracy and efficiency of the method.