The three-dimensional unsteady flow structures around an aerofoil with leading-edge tubercles at angles of attack up to the post-stall regime are experimentally characterised by high-speed three-dimensional particle tracking, physics-informed neural network data assimilation and proper orthogonal decomposition. At mild incidence (${\lesssim}5^{\circ }$), tubercles generate a spanwise-periodic system of counter-rotating vortices, strongly altering the boundary layer from its inception. The vortices adhere to the aerofoil profile, modulating the shear layer by alternating downwash and upwash motions. In the stall regime, the flow features large-scale separation cells spanning several tubercles with no apparent periodicity. Proper orthogonal decomposition reveals a dominant vertical flapping motion of the shear layer with a Strouhal number of $St \approx 0.1$ (based on chord length and free-stream velocity). The second most energetic mode ($St \approx 0.3$) captures part of a streamwise travelling wave, indicating a shedding behaviour. In the deep-stall regime of ${\gtrsim}25^{\circ }$, the unsteadiness is ascribed to the interaction among neighbouring separated patches. The kinematic description of the flow is further utilised to infer the aerodynamic loads through pressure estimation. These observations help understand the gradual attenuation of lift, ascribed to an extended envelope of cellular separation, which contrasts with the abrupt stall observed for straight-edge aerofoils.