  The notion of PRINC domain was introduced by Salce and Zanardo (2014),
motivated by the investigation of the products of idempotent matrices with
entries in a commutative domain. An integral domain R is a PRINC domain if
every two-generated invertible ideal of R is principal. PRINC domains are
closely related to the notion of unique comaximal factorization domain,
introduced by McAdam and Swan (2004). In this article, we prove that there
exist large classes of PRINC domains which are not comaximal factorization
domains, using diverse kinds of constructions. We also produce PRINC domains
that are neither comaximal factorization domains nor projective-free.
