Among the key insights into the glass transition has been the identification of a non-equilibrium phase transition in trajectory space which reveals phase coexistence between the normal supercooled liquid (active phase) and a glassy state (inactive phase). Here we present evidence that such a transition occurs in experiment. In colloidal hard spheres we find a non-Gaussian distribution of trajectories leaning towards those rich in locally favoured structures (LFS), associated with the emergence of slow dynamics. This we interpret as evidence for an non-equilibrium transition to an inactive LFS-rich phase. Reweighting trajectories reveals a first-order phase transition in trajectory space between a normal liquid and a LFS-rich phase. We further find evidence of a purely dynamical transition in trajectory space.