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While data-driven predictive models are a strictly technological construct, they may operate within a social context in which benign engineering choices entail implicit, indirect and unexpected real-life consequences. Fairness of such systems -- pertaining both to individuals and groups -- is one relevant consideration in this space; algorithms can discriminate people across various protected characteristics regardless of whether these properties are included in the data or discernible through proxy variables. To date, this notion has predominantly been studied for a fixed model, often under different classification thresholds, striving to identify and eradicate undesirable, discriminative and possibly unlawful aspects of its operation. Here, we backtrack on this fixed model assumption to propose and explore a novel definition of cross-model fairness where individuals can be harmed when one predictor is chosen ad hoc from a group of equally well performing models, i.e., in view of utility-based model multiplicity. Since a person may be classified differently across models that are otherwise considered equivalent, this individual could argue for a predictor granting them the most favourable outcome, employing which may have adverse effects on other people. We introduce this scenario with a two-dimensional example and linear classification; then, we present a comprehensive empirical study based on real-life predictive models and data sets that are popular with the algorithmic fairness community; finally, we investigate analytical properties of cross-model fairness and its ramifications in a broader context. Our findings suggest that such unfairness can be readily found in real life and it may be difficult to mitigate by technical means alone as doing so is likely to degrade predictive performance.