Some recent Petri net-based approaches to fault diagnosis of distributed systems suggest to factor the problem into local diagnoses based on the unfoldings of local views of the system, which are then correlated with diagnoses from neighbouring supervisors. In this paper we propose a notion of system factorisation expressed in terms of pullback decomposition. To ensure coherence of the local views and completeness of the diagnosis, data exchange among the unfolders needs to be specified with care. We introduce interleaving structures as a format for data exchange between unfolders and we propose a distributed algorithm for computing local views of the unfolding for each system component. The theory of interleaving structures is developed to prove correctness of the distributed unfolding algorithm.