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. In order to gain insight, we investigate the evolution of the mfp in our new massive fully coupled radiation hydrodynamics cosmological simulation of reionization: Cosmic Dawn III (CoDa III). CoDa III's scale ($\rm 94^3 \, cMpc^3$) and resolution ($\rm 8192^3$ grid) make it particularly suitable to study 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 ever RAMSES simulation. A superior agreement with global constraints on reionization is obtained in CoDa III over CoDa II, 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 CoDa III 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 CoDa III is bimodal, with short (neutral) and long (ionized) mfp modes, 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 ~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 $\rm z>13$ ($\rm z<5$).