  Upcoming neutrino telescopes may discover ultra-high-energy (UHE) cosmic
neutrinos, with energies beyond 100 PeV, in the next 10-20 years. Finding their
sources would identify guaranteed sites of interaction of UHE cosmic rays,
whose origin is unknown. We search for sources by looking for multiplets of UHE
neutrinos arriving from similar directions. Our forecasts are state-of-the-art,
geared at neutrino radio-detection in IceCube-Gen2. They account for detector
energy and angular response, and for critical, but uncertain backgrounds.
Sources at declination of $-45^\circ$ to $0^\circ$ will be easiest to discover.
Discovering even one steady-state source in 10 years would imply that the
source has an UHE neutrino luminosity at least larger than about $10^{43}$
erg/s (depending on the source redshift evolution). Discovering no transient
source would disfavor transient sources brighter than $10^{53}$ erg as
dominant. Our results aim to inform the design of upcoming detectors.
