The coherence of an individual quantum state can only be meaningfully discussed when referring to a preferred basis. This arbitrariness can however be lifted when considering sets of quantum states. Here we introduce the concept of set-coherence for characterising the coherence of a set of quantum systems in a basis-independent way. We construct measures for quantifying set-coherence of sets of quantum states as well as quantum measurements. These measures feature an operational meaning in terms of discrimination games and capture precisely the advantage offered by a given set over incoherent ones. Along the way, we also connect the notion of set-coherence to various resource-theoretic approaches recently developed for quantum systems.
