  We develop an optimal Bayesian solution for jointly inferring secondary
signals in the Cosmic Microwave Background (CMB) originating from gravitational
lensing and from patchy screening during the epoch of reionization. This method
is able to extract full information content from the data, improving upon
previously considered quadratic estimators for lensing and screening. We
forecast constraints using the Marginal Unbiased Score Expansion (MUSE) method,
and show that they are largely dominated by CMB polarization, and depend on the
exact details of reionization. For models consistent with current data which
produce the largest screening signals, a detection (3\,$\sigma$) of the
cross-correlation between lensing and screening is possible with SPT-3G, and a
detection of the auto-correlation is possible with CMB-S4. Models with the
lowest screening signals evade the sensitivity of SPT-3G, but are still
possible to detect with CMB-S4 via their lensing cross-correlation.
