This Letter shows that the parametric processes of spontaneous three-photon down-conversion is accompanied by phase tristability of the sub-harmonic signal. The oscillations of the signal in a resonant cavity are modeled through an analytically solvable second-order nonlinear oscillator. Self-sustained oscillations of the signal at a finite amplitude are found to be equally probable in three states with uniform phase contrasts. The onset of oscillations is a case of bifurcation from infinity. The stability of the ternary states is proven through an energy landscape function that identifies the attractor basins of the three states. An analogy is drawn between the oscillation threshold of a three-photon down-conversion oscillator and a first-order phase transition. The investigated phase-tristable oscillator can serve as a classical ternary bit for unconventional computing applications.