When multiple sources provide information about the same unknown quantity, their fusion into a synthetic interpretable message is often a tricky problem, especially when sources are conflicting. In this paper, we propose to use possibility theory and the notion of maximal coherent subsets (MCSs), often used in logic-based representations, to build a fuzzy belief structure that will be instrumental both for extracting useful insight about various features of the information conveyed by the sources and for compressing this information into a unique possibility distribution. Extensions and properties of the basic fusion rule are also studied.