The Poland-Scheraga model describes the denaturation transition of two complementary – in particular, equally long – strands of DNA, and it has enjoyed a remarkable success both for quantitative modeling purposes and at a more theoretical level. The solvable character of the homogeneous version of the model is one of features to which its success is due. In the bio-physical literature a generalization of the model, allowing different length and non complementarity of the strands, has been considered and the solvable character extends to this substantial generalization. We present a mathematical analysis of the homogeneous generalized Poland-Scheraga model. Our approach is based on the fact that such a model is a homogeneous pinning model based on a bivariate renewal process, much like the basic Poland-Scheraga model is a pinning model based on a univariate, i.e. standard, renewal.