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Collective phenomena in systems of interacting agents have helped us understand diverse social, ecological and biological observations. The corresponding explanations are challenged by incorrect information processing. In particular, the models typically assume a shared understanding of signals or a common truth or value system, i.e., an agreement of whether the measurement or perception of information is `right' or `wrong'. It is an open question whether a collective consensus can emerge without these conditions. Here we introduce a model of interacting agents that strive for consensus, however, each with only a subjective perception of the world. Our communication model does not presuppose a definition of right or wrong and the actors can hence not distinguish between correct and incorrect observations. Depending on a single parameter that governs how responsive the agents are to changing their world-view we observe a transition between an unordered phase of individuals that are not able to communicate with each other and a phase of an emerging shared signalling framework. We find that there are two types of convention-aligned clusters: one, where all social actors in the cluster have the same set of conventions, and one, where neighbouring actors have different but compatible conventions (`stable misunderstandings').