  During the first three flights of the Antarctic Impulsive Transient Antenna
(ANITA) experiment, the collaboration detected several neutrino candidates. Two
of these candidate events were consistent with an ultra-high-energy up-going
air shower and compatible with a tau neutrino interpretation. A third neutrino
candidate event was detected in a search for Askaryan radiation in the
Antarctic ice, although it is also consistent with the background expectation.
The inferred emergence angle of the first two events is in tension with IceCube
and ANITA limits on isotropic cosmogenic neutrino fluxes. Here, we test the
hypothesis that these events are astrophysical in origin, possibly caused by a
point source in the reconstructed direction. Given that any ultra-high-energy
tau neutrino flux traversing the Earth should be accompanied by a secondary
flux in the TeV-PeV range, we search for these secondary counterparts in seven
years of IceCube data using three complementary approaches. In the absence of
any significant detection, we set upper limits on the neutrino flux from
potential point sources. We compare these limits to ANITA's sensitivity in the
same direction and show that an astrophysical explanation of these anomalous
events under standard model assumptions is severely constrained regardless of
source spectrum.
