  We study the singular values of the product of two coupled rectangular random
matrices as a determinantal point process. Each of the two factors is given by
a parameter dependent linear combination of two independent, complex Gaussian
random matrices, which is equivalent to a coupling of the two factors via an
Itzykson-Zuber term. We prove that the squared singular values of such a
product form a biorthogonal ensemble and establish its exact solvability. The
parameter dependence allows us to interpolate between the singular value
statistics of the Laguerre ensemble and that of the product of two independent
complex Ginibre ensembles which are both known. We give exact formulae for the
correlation kernel in terms of a complex double contour integral, suitable for
the subsequent asymptotic analysis. In particular, we derive a
Christoffel-Darboux type formula for the correlation kernel, based on a five
term recurrence relation for our biorthogonal functions. It enables us to find
its scaling limit at the origin representing a hard edge. The resulting
limiting kernel coincides with the universal Meijer G-kernel found by several
authors in different ensembles. We show that the central limit theorem holds
for the linear statistics of the singular values and give the limiting variance
explicitly.
