We develop a nonclassicality criterion for the interference of three delayed,
but otherwise identical, light fields in a three-mode Bell interferometer. We
do so by comparing the prediction of quantum mechanics with those of a
classical framework in which independent sources emit electric fields with
random phases. In particular, we evaluate third-order correlations among output intensities as a function of the delays, and show how the presence of a correlation revival for small delays cannot be explained by the classical model of light. The observation of a revival is thus a nonclassicality signature, which can be achieved only by sources with a photon-number statistics that is highly sub-Poissonian. Our analysis provides strong evidence for the nonclassicality of the experiment discussed in [Menssen et al., PRL, 118, 153603 (2017)], and shows how a collective "triad" phase affects the interference of any three or more light fields, irrespective of their quantum or classical character.