  We develop the theory of double multiplicative Poisson vertex algebras. These
structures, defined at the level of associative algebras, are shown to be such
that they induce a classical structure of multiplicative Poisson vertex algebra
on the corresponding representation spaces. Moreover, we prove that they are in
one-to-one correspondence with local lattice double Poisson algebras, a new
important class among Van den Bergh's double Poisson algebras. We derive
several classification results, and we exhibit their relation to non-abelian
integrable differential-difference equations. A rigorous definition of double
multiplicative Poisson vertex algebras in the non-local and rational cases is
also provided.
