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 upgoing 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 7 yr 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.