We present the results of a simulation of the acceptance of the
Antarctic Impulsive Transient Antenna (ANITA) to possible $\nu_\tau$ point source
fluxes detected via $\tau$-lepton-induced air showers. This investigation
is framed around the
detection of four upward-going extensive air shower events observed very close to the horizon in ANITA-IV.
These four events as well as the overall diffuse and point source exposure to Earth-skimming $\nu_\tau$ are also
compared against published ultrahigh-energy neutrino limits from the Pierre Auger Observatory.
We find that while these four events were detected at sky coordinates close to ANITA's maximum $\nu_\tau$ sensitivity
and were not simultaneously visible by Auger, the
implied fluence necessary for ANITA to observe these events is in tension with limits set by Auger across a wide
range of energies and is additionally in tension with ANITA's Askaryan in-ice neutrino channel above~$10^{19}$~eV.