The flavor composition of high-energy astrophysical neutrinos is a rich observable. However, present analyses cannot effectively distinguish particle showers induced by $\nu_e$ versus $\nu_\tau$. We show that this can be accomplished by measuring the intensities of the delayed, collective light emission from muon decays and neutron captures, which are, on average, greater for $\nu_\tau$ than for $\nu_e$. This new technique would significantly improve tests of the nature of astrophysical sources and of neutrino properties. We discuss the promising prospects for implementing it in IceCube and other detectors.