  In this paper, we give formulas that allow one to move between transfer
function type realizations of multi-variate Schur, Herglotz and Pick functions,
without adding additional singularities except perhaps poles coming from the
conformal transformation itself. In the two-variable commutative case, we use a
canonical de Branges-Rovnyak model theory to obtain concrete realizations that
analytically continue through the boundary for inner functions which are
rational in one of the variables (so-called quasi-rational functions). We then
establish a positive solution to McCarthy's Champagne conjecture for local to
global matrix monotonicity in the settings of both two-variable quasi-rational
functions and $d$-variable perspective functions.
