Recent observations of two coherent radio pulses with the ANITA detector can be interpreted as steeply upward-going cosmic-ray showers with energies of a few tenths of an EeV and remain unexplained. The Pierre Auger Observatory has a large exposure to such upward propagating shower-like events, and has used 14 years of its Fluorescence Detector (FD) data to perform a generic search for such events with elevation angles greater than 20$^\circ$ from the horizon. Here this search is recast to constrain models generating high energy $\tau$-leptons. For maximal flexibility, only the propagation, decay, and interactions of $\tau$-leptons are treated in this analysis, meaning that the results are independent of the $\tau$-production scenario. This treatment allows for the application of these results to the wide range of models producing $\tau$-leptons that have been proposed to describe the "anomalous" ANITA events. The goal of this study is accomplished by generating $\tau$-leptons within the Earth and its atmosphere with an intensity dependent on the media density. The zenith angle, location and calorimetric energy of any resulting $\tau$-induced air showers are then used to calculate the exposure of the FD of the Pierre Auger Observatory to $\tau$ primaries. Differential limits as low as $10^{-9}$GeV s$^{-1}$cm$^{-2}$sr$^{-1}$ to the flux of $\tau$-leptons produced with less than a 50 km path length below the Earth's surface are reported for several zenith angle ranges and primary energy spectra. Full exposure and sensitivity information is provided, facilitating the application of these results to different $\tau$-lepton production models.