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.
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