  We consider the implications of Lorentz-invariance violation (LIV) on
cosmogenic neutrino observations, with particular focus on the constraints
imposed on several well-developed models for ultra-high energy cosmogenic
neutrino production by recent results from the Antarctic Impulsive Transient
Antenna (ANITA) long-duration balloon payload, and Radio Ice Cherenkov
Experiment (RICE) at the South Pole. Under a scenario proposed originally by
Coleman and Glashow, each lepton family may attain maximum velocities that can
exceed the speed of light, leading to energy-loss through several interaction
channels during propagation. We show that future observations of cosmogenic
neutrinos will provide by far the most stringent limit on LIV in the neutrino
sector. We derive the implied level of LIV required to suppress observation of
predicted fluxes from several mainstream cosmogenic neutrino models, and
specifically those recently constrained by the ANITA and RICE experiments. We
simulate via detailed Monte Carlo code the propagation of cosmogenic neutrino
fluxes in the presence of LIV-induced energy losses. We show that this process
produces several detectable effects in the resulting attenuated neutrino
spectra, even at LIV-induced neutrino superluminality of (u_{\nu}-c)/c ~
10^{-26}, about 13 orders of magnitude below current bounds.
