  Recent measurements of the ionizing photon mean free path
($\lambda_{912}^{\rm mfp}$) at $5 < z < 6$ suggest that the IGM was rapidly
evolving at those times. We use radiative transfer simulations to explore the
implications for reionization, with a focus on the short value of
$\lambda_{912}^{\rm mfp} = 3.57^{+3.09}_{-2.14}$ cMpc/$h$ at $z=6$. We
introduce a model for sub-resolution ionizing photon sinks based on radiative
hydrodynamics simulations of small-scale IGM clumping. We argue that the rapid
evolution in $\lambda_{912}^{\rm mfp}$ at $z=5-6$, together with constraints on
the metagalactic ionizing background, favors a late reionization process in
which the neutral fraction evolved rapidly in the latter half. We also argue
that the short $\lambda_{912}^{\rm mfp}(z=6)$ points to faint galaxies as the
primary drivers of reionizaton. Our preferred model, with $\lambda_{912}^{\rm
mfp}(z=6) = 6.5$ Mpc/$h$, has a midpoint of $z= 7.1$ and ends at $z= 5.1$. It
requires 3 ionizing photons per H atom to complete reionization and a LyC
photon production efficiency of $\log(f^{\rm eff}_{\rm esc} \xi_{\rm
ion}/[\mathrm{erg}^{-1} \mathrm{Hz}]) = 24.8$ at $z>6$. Recovering
$\lambda_{912}^{\rm mfp}(z=6)$ as low as the measured central value may require
an increase in IGM clumpiness beyond predictions from simulations, with a
commensurate increase in the photon budget.
