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The cross-correlation between fluctuations in the electron scattering opticaldepth $\tau_{\rm es}$ as probed by future Cosmic Microwave Background (CMB)experiments, and fluctuations in the 21cm differential brightness temperature$\Delta T_{\rm 21cm}$ as probed by ground-based radio interferometers, willtrace the reionization history of the Universe. In particular, the $\tau_{\rmes}-$21cm cross-correlation should yield a determination of the characteristicbubble size distribution and ionization fraction as a function of redshift.When assuming that the cross-correlation signal is limited by instrumentalnoise rather than by foregrounds, we estimate its potential detectability byupcoming experiments. Specifically, the combination of HERA and SimonsObservatory, CMB-S4 and PICO should yield a signal-to-noise ratio around 3 - 6,while and the exploitation of the SKA should increase it to 10-20. Finally, wehave discussed how such levels of detectability can be affected when (simplymodeled) 21cm foregrounds are present. For the most promising PICO$\times$SKAconfiguration, an efficiency of foreground removal to a level of $7\times10^{-4}$ is needed to achieve a $5\sigma$ detection of the cross-correlationsignal; in addition, safe avoidance of foreground contamination in theline-of-sight Fourier modes above $0.03 \,h\,\rm Mpc^{-1}$ would guarantee adetection significance around $3\sigma$.

Detectability of the $\tau_{es}$-21cm cross-correlation: a tomographic probe of patchy reionization