  Ultra-high-energy (UHE) cosmic neutrinos, with energies above 100 PeV, could
be finally discovered in the near future. Measuring their flavor composition
would reveal information about their production and propagation, but new
techniques are needed for UHE neutrino telescopes to have the capabilities to
do it. We address this by proposing a new way to measure the UHE neutrino
flavor composition that does not rely on individual telescopes having
flavor-identification capabilities. We manufacture flavor sensitivity from the
joint detection by one telescope sensitive to all flavors -- the radio array of
IceCube-Gen2 -- and one mostly sensitive to $\nu_\tau$ -- GRAND. From this
limited flavor sensitivity, predominantly to $\nu_\tau$, and even under
conservative choices of neutrino flux and detector size, we extract important
insight. For astrophysics, we forecast meaningful constraints on the neutrino
production mechanism; for fundamental physics, vastly improved constraints on
Lorentz-invariance violation. These are the first measurement forecasts of the
UHE $\nu_\tau$ content.
