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' behavior 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 threephoton interference by generating heralded single photons and interfering them in a fiber tritter. Using multiple degrees of freedom-temporal delays and polarization-we isolate three-photon interference from two-photon interference. Our experiment disproves the view that pairwise two-photon distinguishability uniquely determines the degree of nonclassical many-particle interference. DOI: 10.1103/PhysRevLett.118.153603 The famous Hong-Ou-Mandel (HOM) experiment in 1987 provided the first important example of nonclassical two-photon interference [1]. Two independent photons impinging on a beam splitter exhibit bunching behavior at the output ports that cannot be explained by a classical field model. The degree of bunching depends on how similar the two photons are in all degrees of freedom, for example, time, frequency, polarization, and spatial mode. Extending the study of interference to many particles is of interest from a fundamental as well as from a technological viewpoint [2][3][4][5][6][7]. The scattering of multiple photons in linear networks is related to solving problems in quantum information processing, metrology, and quantum state engineering [8][9][10][11][12][13][14][15][16]. Thus, understanding multiphoton interference is also of great relevance for practical applications.Here, we demonstrate how many-particle interference is fundamentally richer than two-particle interference [17]. Two situations with the same pairwise distinguishability can lead to a different output distribution. This is due to a phase, the triad phase, that occurs only when more than two photons interfere.We use independent photons and a tritter, a three-port symmetric beam splitter to investigate many-particle interference. We isolate the triad phase for the first time by interfering three photons in a tritter and exploiting multiple degrees of freedom, here time and polarization. We show that interfering three identical photons and varying time delays between them, as demonstrated in previous work [5,18,19], is not sufficient to study three-photon interference in full generality [20,21]. Our experiment allows us to isolate and tune the three-photon interference term as distinct from two-photon interference. In particular, manipulation of the triad phase goes beyond what is possible using temporal...