  We consider a non-Hermitian matrix orthogonality on a contour in the complex
plane. Given a diagonalizable and rational matrix valued weight, we show that
the Christoffel--Darboux (CD) kernel, which is built in terms of matrix
orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of
meromorphic functions in a Riemann surface. If this Riemann surface has genus
$0$, then the matrix valued CD kernel is equivalent to a scalar reproducing
kernel of polynomials in the plane. Interestingly, this scalar reproducing
kernel is not necessarily a scalar CD kernel. As an application of our result,
we show that the correlation kernel of certain doubly periodic lozenge tiling
models admits a double contour integral representation involving only a scalar
CD kernel. This simplifies a formula of Duits and Kuijlaars.
