  Bound states in superconductors are expected to exhibit a spatially resolved
electron-hole asymmetry which is the hallmark of their quantum nature. This
asymmetry manifests as oscillations at the Fermi wavelength, which is usually
tiny and thus washed out by thermal broadening or by scattering at defects.
Here we demonstrate theoretically and confirm experimentally that, when coupled
to magnetic impurities, bound states in a vortex core exhibit an emergent axial
electron-hole asymmetry on a much longer scale, set by the coherence length. We
study vortices in 2H-NbSe$_2$ and in 2H-NbSe$_{1.8}$S_{0.2}$ with magnetic
impurities, characterizing these with detailed Hubbard-corrected density
functional calculations. We find that the induced electron-hole imbalance
depends on the band character of the superconducting material. Our results open
interesting prospects for the study of coupled superconducting bound states.
