  The recent observation of NGC 1068 by the IceCube Neutrino Observatory has
opened a new window to neutrino physics with astrophysical baselines. In this
Letter, we propose a new method to probe the nature of neutrino masses using
these observations. In particular, our method enables searching for signatures
of pseudo-Dirac neutrinos with mass-squared differences that reach down to
$\delta m^2 \gtrsim 10^{-21}~\text{eV}^2$, improving the reach of terrestrial
experiments by more than a billion. Finally, we discuss how the discovery of a
constellation of neutrino sources can further increase the sensitivity and
cover a wider range of $\delta m^2$ values.
