We calculate the hydrogen and helium-ionizing radiation escaping star-forming molecular clouds, as a function of the star cluster mass and compactness, using a set of high-resolution radiation-magneto-hydrodynamic simulations of star formation in self-gravitating, turbulent molecular clouds. In these simulations, presented in He, Ricotti and Geen (2019), the formation of individual massive stars are well resolved, and their UV radiation feedback and lifetime on the main sequence are modeled self-consistently. We find that the escape fraction of ionizing radiation from molecular clouds, $langle f_{rm esc}^{scriptscriptstyle rm MC}rangle$, decreases with increasing mass of the star cluster and with decreasing compactness. Molecular clouds with densities typically found in the local Universe have negligible $langle f_{rm esc}^{scriptscriptstyle rm MC}rangle$, ranging between $0.5%$ to $5%$. Ten times denser molecular clouds have $langle f_{rm esc}^{scriptscriptstyle rm MC}rangle approx 10%-20%$, while $100times$ denser clouds, which produce globular cluster progenitors, have $langle f_{rm esc}^{scriptscriptstyle rm MC}rangle approx 20%-60%$. We find that $langle f_{rm esc}^{scriptscriptstyle rm MC}rangle$ increases with decreasing gas metallicity, even when ignoring dust extinction, due to stronger radiation feedback. However, the total number of escaping ionizing photons decreases with decreasing metallicity because the star formation efficiency is reduced. We conclude that the sources of reionization at $z>6$ must have been very compact star clusters forming in molecular clouds about $100times$ denser than in todays Universe, which leads to a significant production of old globular clusters progenitors.
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