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A two-fold is a singular point on the discontinuity surface of apiecewise-smooth vector field, at which the vector field is tangent to thediscontinuity surface on both sides. If an orbit passes through an invisibletwo-fold (also known as a Teixeira singularity) before settling to regularperiodic motion, then the phase of that motion cannot be determined frominitial conditions, and in the presence of small noise the asymptotic phase ofa large number of sample solutions is highly random. In this paper we show howthe probability distribution of the asymptotic phase depends on the globalnonlinear dynamics. We also show how the phase of a smooth oscillator can berandomised by applying a simple discontinuous control law that generates aninvisible two-fold. We propose that such a control law can be used todesynchronise a collection of oscillators, and that this manner of phaserandomisation is fast compared to existing methods (which use fixed points asphase singularities) because there is no slowing of the dynamics near atwo-fold.