  We perform extensive numerical simulations of a paradigmatic model glass
former, the hard-sphere fluid with 10% polydispersity. We sample from the
ensemble of trajectories with fixed observation time, whereby single
trajectories are generated by event-driven molecular dynamics. We show that
these trajectories can be characterized in terms of local structure, and we
find a dynamical-structural (active-inactive) phase transition between two
dynamical phases: one dominated by liquid-like trajectories with low degree of
local order and one dominated by glassy-like trajectories with a high degree of
local order. We show that both phases coexist and are separated by a
spatiotemporal interface. Sampling exceptionally long trajectories allows to
perform a systematic finite-size scaling analysis. We find excellent agreement
with Binder's scaling theory for first-order transitions. Interestingly, the
coexistence region narrows at higher densities, supporting the idea of a
critical point controlling the dynamic arrest.
