  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.
