Recently, the ANITA collaboration reported on two upward-going extensive air shower events consistent with a primary particle that emerges from the surface of the ice. These events may be of $\nu_\tau$ origin, in which the neutrino interacts within the Earth to produce a $\tau$ lepton that emerges from the Earth, decays in the atmosphere, and initiates an extensive air shower. In this paper we estimate an upper bound on the ANITA acceptance to a diffuse $\nu_\tau$ flux detected via $\tau$-lepton-induced air showers within the bounds of Standard Model (SM) uncertainties. By comparing this estimate with the acceptance of Pierre Auger Observatory and IceCube and assuming SM interactions, we conclude that a $\nu_\tau$ origin of these events would imply a neutrino flux at least two orders of magnitude above current bounds. Comment: 12 pages, 7 figures
