  Determining the electron neutrino mass by electron capture in $^{163}$Ho
relies on an accurate understanding of the differential electron capture
nuclear decay rate as a function of the distribution of the total decay energy
between the neutrino and electronic excitations. The resulting spectrum is
dominated by resonances due to local atomic multiplet states with core holes.
Coulomb scattering between electrons couple the discrete atomic states, via
Auger-Meitner decay, to final states with free electrons. The atomic multiplets
are above the auto-ionisation energy, such that the delta functions
representing these discrete levels turn into a superposition of Lorentzian,
Mahan- and Fano-like line-shapes. We present an \textit{ab initio} method to
calculate nuclear decay modifications due to such processes. It includes states
with multiple correlated holes in local atomic orbitals interacting with
unbound Auger-Meitner electrons. A strong energy-dependent, asymmetric
broadening of the resonances in good agreement with recent experiments is
found. We present a detailed analysis of the mechanisms determining the final
spectral line-shape and discuss both the Fano interference between different
resonances, as well as the energy dependence of the Auger-Meitner Coulomb
matrix elements. The latter mechanism is shown to be the dominant channel
responsible for the asymmetric line-shape of the resonances in the electron
capture spectrum of $^{163}$Ho.
