  The two-matrix model is defined on pairs of Hermitian matrices $(M_1,M_2)$ of
size $n\times n$ by the probability measure $$\frac{1}{Z_n}
\exp\left(\textrm{Tr} (-V(M_1)-W(M_2)+\tau M_1M_2)\right)\ dM_1\ dM_2, $$ where
$V$ and $W$ are given potential functions and $\tau\in\er$. We study averages
of products and ratios of characteristic polynomials in the two-matrix model,
where both matrices $M_1$ and $M_2$ may appear in a combined way in both
numerator and denominator. We obtain determinantal expressions for such
averages. The determinants are constructed from several building blocks: the
biorthogonal polynomials $p_n(x)$ and $q_n(y)$ associated to the two-matrix
model; certain transformed functions $\P_n(w)$ and $\Q_n(v)$; and finally
Cauchy-type transforms of the four Eynard-Mehta kernels $K_{1,1}$, $K_{1,2}$,
$K_{2,1}$ and $K_{2,2}$. In this way we generalize known results for the
$1$-matrix model. Our results also imply a new proof of the Eynard-Mehta
theorem for correlation functions in the two-matrix model, and they lead to a
generating function for averages of products of traces.
