A *-superalgebra is an associative \(F\)-algebra (where \(F\) is a field) endowed with a \(\mathbb{Z}/2\mathbb{Z}\)-grading and with a \(\mathbb{Z}/2\mathbb{Z}\)-graded involution *. Let \(A\) be a finitely generated *-superalgebra satisfying a *-graded polynomial identity, and assume that \(F\) has characteristic zero. The *-graded exponent (of the *-graded variety generated by \(A\)) is a non-negative integer measuring the growth of the codimensions of the spaces of multilinear *-graded polynomial identities satisfied by \(A\). The *-graded variety generated by \(A\) is minimal if any of its proper subvarieties has strictly smaller *-graded exponent. In an earlier paper the same authors introduced a construction of a *-superalgebra \(A\) (a subalgebra of a block upper triangular matrix algebra associated with a sequence \(A_1,\dots,A_m\) of simple algebras), such that the minimal *-graded varieties are exactly the varieties generated by these algebras. In the present paper they characterize the cases when the ideal of *-graded identities of \(A\) has a factorization in terms of the ideals of *-graded identities of the simple algebras \(A_i\), analogously to the corresponding theory for ordinary PI-algebras.