It was recently suggested that a novel type of phase transition may occur in the visibility of electronic Mach-Zehnder interferometers. Here, we present experimental evidence for the existence of this transition. The transition is induced by strongly non-Gaussian noise that originates from the strong coupling of a quantum point contact to the interferometer. We provide a transparent physical picture of the effect by exploiting a close analogy to the neutrino oscillations of particle physics. In addition, our experiment constitutes a probe of the singularity of the elusive full counting statistics of a quantum point contact.