We estimate the rate of observable horizontal and upward τ air showers (HORTAUs and UPTAUs, respectively) considering both the Earth's opacity and the contribution of the terrestrial atmosphere. Our result applies to most neutrino telescope projects, especially to the Extreme Universe Space Observatory (EUSO). Using a compact analytical formula, we calculate the effective target volumes and masses for τ air showers emerging from the Earth. The resulting model-independent effective masses for EUSO may encompass-at Eντ~=1019 eV-a huge average volume (~=1020 km3) compared to current neutrino experiments. Adopting simple power-law neutrino fluxes, dNν/dEν~E-2 and E-1, calibrated to Greisen-Zapetin-Kuzmin (GZK) like and Z-burst-like models, we estimate that at E~=1019 eV, nearly half a dozen horizontal shower events should be detected by EUSO in 3 yr of data collection, considering the 10% duty cycle efficiency and a minimal τ neutrino energy fluence φνEν~=50 eV cm-2 s-1 sr-1. The detection of HORTAUs may test the ``guaranteed'' GZK neutrino flux (secondaries of photopion production due to ultra-high-energy cosmic-ray [UHECR] scattering onto 2.75 K cosmic background radiation). We also find that the equivalent mass for an outer layer made of rock is larger than the water mass, contrary to simplified all-rock/all-water Earth models and previous studies. Therefore, we expect an enhancement of neutrino detection along continental shelves near the highest mountain ranges, because of the better geometrical acceptance of Earth-skimming neutrinos. In this picture, the Auger experiment might reveal such an increase at Eν~=1018 eV (with 26 events in 3 yr) if the angular resolution (both in azimuth and zenith) reaches an accuracy of nearly 1° (or below); the angular accuracy depends on the the morphology of the Andes, the distance of each element of the array from the mountain range, and the local terrestrial magnetic fields. Such a high angular resolution is necessary to disentangle τ air showers from more abundant downward horizontal UHECRs. Finally, we show that the number of UHE-induced τ air-shower events is larger at energies lower than Eν~1020 or Eν~1019 eV; therefore we suggest an extension of the EUSO sensitivity at the lowest possible energy.