Recent determinations of the mean free path of ionising photons (mfp) in the intergalactic medium (IGM) at $\rm z=6$ are lower than many theoretical predictions. To gain insight into this issue, we investigate the evolution of the mfp in our new massive fully coupled radiation hydrodynamics cosmological simulation of reionization: Cosmic Dawn III (CoDaIII). CoDaIII's scale ($\rm 94^3 \, cMpc^3$) and resolution ($\rm 8192^3$ grid) make it particularly suitable to study the evolution of the IGM during Reionization. The simulation was performed with RAMSES-CUDATON on Summit, and used 131072 processors coupled to 24576 GPUs, making it the largest Reionization simulation, and largest RAMSES simulation ever performed. A superior agreement with global constraints on Reionization is obtained in CoDaIII over CoDaII especially for the evolution of the neutral hydrogen fraction and the cosmic photo-ionization rate, thanks to an improved calibration, later end of reionization ($\rm z=5.6$), and higher spatial resolution. Analyzing the mfp, we find that CoDaIII reproduces the most recent observations very well, from $\rm z=6$ to $\rm z=4.6$. We show that the distribution of the mfp in CoDaIII is bimodal, with short (neutral) and long (ionized) mfp modes, respectively, due to the patchiness of reionization and the co-existence of neutral versus ionized regions during Reionization. The neutral mode peaks at sub-kpc to kpc scales of mfp, while the ionized mode peak evolves from $\rm 0.1 Mpc/h$ at $\rm z=7$ to $\sim 10$ Mpc/h at $\rm z=5.2$. Computing the mfp as the average of the ionized mode provides the best match to the recent observational determinations. The distribution reduces to a single neutral (ionized) mode at $\rm z>13$ ($\rm z<5$).