  We show a number of Toda brackets in the homotopy of the motivic bordism
spectrum $MGL$ and of the Real bordism spectrum $MU_{\mathbb R}$. These
brackets are "red-shifting" in the sense that while the terms in the bracket
will be of some chromatic height $n$, the bracket itself will be of chromatic
height $(n+1)$. Using these, we deduce a family of exotic multiplications in
the $\pi_{(\ast,\ast)}MGL$-module structure of the motivic Morava $K$-theories,
including non-trivial multiplications by $2$. These in turn imply the analogous
family of exotic multiplications in the $\pi_{\star}MU_\mathbb R$-module
structure on the Real Morava $K$-theories.
