I will present the status of our pioneering pipeline to robustly and self-consistently extract the sky-averaged signal of primordial neutral hydrogen from a foreground that is 4-6 orders of magnitude larger at low radio frequencies and weighted by a chromatic beam. The pipeline utilizes pattern recognition and Bayesian inference, properly accounting for the covariance between signal and beam-weighted foreground. After obtaining optimal basis vectors from training sets of these two components, it selects a minimal number of terms to best separate them via a linear, simultaneous fit. Utilizing the corresponding spectral constraints as starting point for a Markov Chain Monte Carlo (MCMC) algorithm, any chosen nonlinear signal model can be sampled without traversing the beam-weighted foreground parameter space. This is performed by analytically marginalizing over the latter, linear space at each step of the MCMC, which drastically augments the efficiency of the exploration. In addition, this allows us to straightforwardly utilize a sufficiently large number of beam-weighted foreground modes, of which we suppress the unimportant modes by applying priors derived from the training set, avoiding the need for a precise selection of the number of modes to be marginalized over. Hence, our technique is able to incorporate increasingly complex descriptions of the beam-weighted foreground model with negligible computational time costs. The ability to marginalize over a large number of linear parameters to capture key differences between the foreground and the unpolarized, isotropic signal increases the separation power between them. Using simulations, we show that such beam-weighted foreground model complexity encoding differential features from the signal is crucial for accurate and precise fits. Correlating multiple sky views and polarization (using the four Stokes parameters) foreground measurements significantly helps in separating the global 21-cm signal, unpolarized and unchanged by sky view variations, and the beam-weighted foreground, modified by those changes. Our general pipeline can be applied to available data sets such as those from the Experiment to Detect the Global EoR (Epoch of Reionization) Signature (EDGES), but is particularly built for NASA’s Dark Ages Polarimeter PathfindER (DAPPER) mission concept, and its ground-based prototype, the Cosmic Twilight Polarimeter (CTP).