  Dark matter searches using dual-phase xenon time-projection chambers
(LXe-TPCs) rely on their ability to reject background electron recoils (ERs)
while searching for signal-like nuclear recoils (NRs). ER response is typically
calibrated using $\beta$-decay sources, such as tritium, but these calibrations
do not characterize events accompanied by an atomic vacancy, as in solar
neutrino scatters off inner shell electrons. Such events lead to emission of
X-rays and Auger electrons, resulting in higher electron-ion recombination and
thus a more NR-like response than inferred from $\beta$-decay calibration. We
present a cross-calibration of tritium $\beta$-decays and $^{127}$Xe
electron-capture decays (which produce inner-shell vacancies) in a small-scale
LXe-TPC and give the most precise measurements to date of light and charge
yields for the $^{127}$Xe L-shell electron-capture in liquid xenon. We observe
a 6.9$\sigma$ (9.2$\sigma$) discrepancy in the L-shell capture response
relative to tritium $\beta$-decays, measured at a drift field of 363 $\pm$ 14
V/cm (258 $\pm$ 13 V/cm), when compared to simulations tuned to reproduce the
correct $\beta$-decay response. In dark matter searches, use of a background
model that neglects this effect leads to overcoverage (higher limits) for
background-only multi-kiloton-year exposures, but at a level much less than the
1-$\sigma$ experiment-to-experiment variation of the 90\% C.L. upper limit on
the interaction rate of a 50 GeV/$c^2$ dark matter particle.
