  We have obtained spectra of 163 quasars at $z_\mathrm{em}>4.4$ with the
Gemini Multi Object Spectrometers on the Gemini North and South telescopes, the
largest publicly available sample of high-quality, low-resolution spectra at
these redshifts. From this homogeneous data set, we generated stacked quasar
spectra in three redshift intervals at $z\sim 5$. We have modelled the flux
below the rest-frame Lyman limit ($\lambda_\mathrm{r}<912$\AA) to assess the
mean free path $\lambda_\mathrm{mfp}^{912}$ of the intergalactic medium to
HI-ionizing radiation. At mean redshifts $z_\mathrm{q}=4.56$, 4.86 and 5.16, we
measure $\lambda_\mathrm{mfp}^{912}=(22.2\pm 2.3, 15.1\pm 1.8, 10.3\pm
1.6)h_{70}^{-1}$ proper Mpc with uncertainties dominated by sample variance.
Combining our results with $\lambda_\mathrm{mfp}^{912}$ measurements from lower
redshifts, the data are well modelled by a simple power-law
$\lambda_\mathrm{mfp}^{912}=A[(1+z)/5]^\eta$ with $A=(37\pm 2)h_{70}^{-1}$ Mpc
and $\eta = -5.4\pm 0.4$ between $z=2.3$ and $z=5.5$. This rapid evolution
requires a physical mechanism -- beyond cosmological expansion -- which reduces
the cosmic effective Lyman limit opacity. We speculate that the majority of HI
Lyman limit opacity manifests in gas outside galactic dark matter haloes,
tracing large-scale structures (e.g. filaments) whose average density (and
consequently neutral fraction) decreases with cosmic time. Our measurements of
the strongly redshift-dependent mean free path shortly after the completion of
HI reionization serve as a valuable boundary condition for numerical models
thereof. Having measured $\lambda_\mathrm{mfp}^{912}\approx 10$ Mpc at $z=5.2$,
we confirm that the intergalactic medium is highly ionized by that epoch and
that the redshift evolution of the mean free path does not show a break that
would indicate a recent end to HI reionization.
