  The reionization epoch concludes when ionizing photons reach every corner of
the Universe. Reionization has generally been assumed to be limited primarily
by the rate at which galaxies produce ionizing photons, but the recent
measurement of a surprisingly short ionizing photon mean free path of
$0.75^{+0.65}_{-0.45}$ proper Mpc at $z = 6$ by Becker et al. (2021) suggests
that absorption by residual neutral hydrogen in the otherwise ionized
intergalactic medium may play a much larger role than previously expected. Here
we show that consistency between this short mean free path and the coeval dark
pixel fraction in the Ly$\alpha$ forest requires a cumulative output of
$6.1^{+11}_{-2.4}$ ionizing photons per baryon by reionization's end. This
represents a dramatic increase in the ionizing photon budget over previous
estimates, greatly exacerbating the tension with measurements of the ionizing
output from galaxies at later times. Translating this constraint into the
instantaneous ionizing production from galaxies in our model, we find
$\log_{10}f_{\rm esc}\xi_{\rm ion}/\text{(erg/Hz)}^{-1} =25.02_{-0.21}^{+0.45}$
at $z\sim6$. Even with optimistic assumptions about the ionizing production
efficiency of early stellar populations, and assuming the galaxy luminosity
function extends to extremely faint sources ($M_{\text{UV}}\leq-11$), complete
reionization requires the escape fraction of ionizing photons to exceed $20\%$
across the galaxy population. This is far larger than observed in any galaxy
population at lower redshifts, requiring rapid evolution in galaxy properties
after the first billion years of cosmic time. This tension cannot be completely
relieved within existing observational constraints on the hydrogen neutral
fraction and mean free path.
