In nonconvex optimisation, commonly used methods are usually shown to converge to stationary points, while guarantees of convergence to local minimisers have comparatively received less attention. To close this gap, we analyse the convergence behaviour of the Davis–Yin three-operator splitting method. More precisely, for structured weakly convex semialgebraic optimisation problems, we establish convergence to local minimisers, almost surely. We propose a novel perspective based on the so-called shadow sequence, deviating from the traditional strategy that uses fixed-point sequences. We perform numerical experiments that suggest that the Davis–Yin method is more robust with respect to the selection of parameters than the forward-backward method, even in settings where the latter may seem to be the obvious choice for solution method.