  We investigate spectral properties of a Hermitised random matrix product
which, contrary to previous product ensembles, allows for eigenvalues on the
full real line. We prove that the eigenvalues form a bi-orthogonal ensemble,
which reduces asymptotically to the Hermite Muttalib-Borodin ensemble. Explicit
expressions for the bi-orthogonal functions as well as the correlation kernel
are provided. Scaling the latter near the origin gives a limiting kernel
involving Meijer G-functions, and the functional form of the global density is
calculated. As a part of this study, we introduce a new matrix transformation
which maps the space of polynomial ensembles onto itself. This matrix
transformation is closely related to the so-called hyperbolic
Harish-Chandra-Itzykson-Zuber integral.
