The ANtarctic Impulsive Transinet Antenna (ANITA) recently reported two $\sim$EeV showers which, at first glance, could be interpreted as emerging tau neutrinos. Their emergence angle was shown to be in tension with the expected angular distribution from isotropic astrophysical neutrinos. However, localized emission from a point source in the event's direction can evade diffuse bounds. In this work, we present an updated calculation of the tau-regeneration effect using a newly developed Monte Carlo package, \TauRunner. We describe the algorithm and the software\’s capabilities, and apply it to the anomalous ANITA detections. We show that any flux from the direction of the ANITA events should be accompanied by secondary tau-neutrinos detectable at IceCube. We derive the maximum allowed secondary flux at IceCube and compare it to the flux required to detect one event at ANITA. We find that these fluxes are in severe tension, requiring a significant over-fluctuation to account for the anomalous event in 2014.