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 $zsim 5$. 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 $lambda_mathrm{mfp}^{912}=(22.2pm 2.3, 15.1pm 1.8, 10.3pm 1.6)h_{70}^{-1}$ 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 $A=(37pm 2)h_{70}^{-1}$ Mpc and $eta = -5.4pm 0.4$ 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.
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