  We analyze in detail the angular distributions in $\bar{B}\to D^*\ell
\bar\nu$ decays, with a focus on lepton-flavour non-universality. We
investigate the minimal number of angular observables that fully describes
current and upcoming datasets, and explore their sensitivity to physics beyond
the Standard Model (BSM) in the most general weak effective theory. We apply
our findings to the current datasets, extract the non-redundant set of angular
observables from the data, and compare to precise SM predictions that include
lepton-flavour universality violating mass effects. Our analysis shows that the
current presentation of the experimental data is not ideal and prohibits the
extraction of the full set of relevant BSM parameters, since the number of
independent angular observables that can be inferred from data is limited to
only four. We uncover a $\sim4\sigma$ tension between data and predictions that
is hidden in the redundant presentation of the Belle 2018 data on $\bar{B}\to
D^*\ell \bar\nu$ decays. This tension specifically involves observables that
probe $e-\mu$ lepton-flavour universality. However, we find inconsistencies in
these data, which renders results based on it suspicious. Nevertheless, we
discuss which generic BSM scenarios could explain the tension, in the case that
the inconsistencies do not affect the data materially. Our findings highlight
that $e-\mu$ non-universality in the SM, introduced by the finite muon mass, is
already significant in a subset of angular observables with respect to the
experimental precision.

High Energy Physics - Phenomenology;High Energy Physics - Experiment