The deviations with respect to the Standard Model (SM) that are currently observed in b → sll transitions (the so-called flavour anomalies) can be interpreted in terms of different New Physics (NP) scenarios within a model-independent effective approach. We reconsider the determination of NP in global fits from a different perspective by removing one implicit hypothesis of current analyses, namely that NP is only Lepton-Flavour Universality Violating (LFUV). We examine the roles played by LFUV NP and Lepton-Flavour Universal (LFU) NP altogether, providing new directions to identify the possible theory beyond the SM responsible for the anomalies observed. New patterns of NP emerge due to the possibility of allowing at the same time large LFUV and LFU NP contributions to C10μ, which provides a different mechanism to obey the constraint from the Bs → μ+μ− branching ratio. In this landscape of NP, we discuss how to discriminate among these scenarios in the short term thanks to current and forthcoming observables. While the update of RK will be a major milestone to confirm the NP origin of the flavour anomalies, additional observables, and in particular the LFUV angular observable Q5, turn out to be central to assess the precise NP scenario responsible for the observed anomalies.