  Performing both right and left multiplication operations using general
regular matrix polynomials, which need not be monic and may possess leading
coefficients of arbitrary rank, on a rectangular matrix of measures associated
with mixed multiple orthogonal polynomials, reveals corresponding Christoffel
formulas. These formulas express the perturbed mixed multiple orthogonal
polynomials in relation to the original ones. Utilizing the divisibility
theorem for matrix polynomials, we establish a criterion for the existence of
perturbed orthogonality, expressed through the non-cancellation of certain
$\tau$ determinants.
