The Poland-Scheraga model for DNA denaturation, besides playing a central role in applications, has been widely studied in the physical and mathematical literature over the past decades. More recently a natural generalization has been introduced in the biophysics literature to overcome the limits of the original model, namely to allow an excess of bases -- i.e. a different length of the two single stranded DNA chains -- and to allow slippages in the chain pairing. The increased complexity of the model is reflected in the appearance of configurational transitions when the DNA is in double stranded form. In a previous work of two of the authors the generalized Poland-Scheraga model has been analyzed thanks to a representation in terms of a bivariate renewal process. In this work we exploit this representation farther and fully characterize the path properties of the system, making therefore explicit the geometric structures -- and the configurational transitions -- that are observed when the polymer is in the double stranded form. What we prove is that, when the excess of bases is not absorbed in a homogeneous fashion along the double stranded chain, then it either condensates in a single macroscopic loop or it accumulates into an unbound single strand free end.