Advancing over up-to-date information theoretic results that assume symmetric correlation models, in this work we consider the problem of lossy binary source coding with side information, where the correlation is expressed by a generic binary asymmetric channel. Specifically, we present an in-depth analysis of rate distortion with side information available to both the encoder and decoder (conventional predictive), as well as the Wyner-Ziv problem for this particular setup. Prompted by our recent results for the Z-channel correlation case, we evaluate the rate loss between the Wyner-Ziv and the conventional predictive coding, as a function of the parameters of the binary asymmetric correlation channel.