  A $\theta$-deformation of the Laguerre weighted Cauchy two-matrix model, and
the Bures ensemble, is introduced. Such a deformation is familiar from the
Muttalib-Borodin ensemble. The $\theta$-deformed Cauchy-Laguerre two-matrix
model is a two-component determinantal point process. It is shown that the
correlation kernel, and its hard edge scaled limit, can be written as the Fox
H-functions, generalising the Meijer G-function class known from the study of
the case $\theta= 1$. In the $\theta=1$ case, it is shown Laguerre-Bures
ensemble is related to the Laguerre-Cauchy two-matrix model, notwithstanding
the Bures ensemble corresponds to a Pfaffian point process. This carries over
to the $\theta$-deformed case, allowing explicit expressions involving Fox
H-functions for the correlation kernel, and its hard edge scaling limit, to be
obtained.
