  Quantum interference of two independent particles in pure quantum states is
fully described by the particles' distinguishability: the closer the particles
are to being identical, the higher the degree of quantum interference. When
more than two particles are involved, the situation becomes more complex and
interference capability extends beyond pairwise distinguishability, taking on a
surprisingly rich character. Here, we study many-particle interference using
three photons. We show that the distinguishability between pairs of photons is
not sufficient to fully describe the photons' behaviour in a scattering
process, but that a collective phase, the triad phase, plays a role. We are
able to explore the full parameter space of three-photon interference by
generating heralded single photons and interfering them in a fibre tritter.
Using multiple degrees of freedom - temporal delays and polarisation - we
isolate three-photon interference from two-photon interference. Our experiment
disproves the view that pairwise two-photon distinguishability uniquely
determines the degree of non-classical many-particle interference.
