  We construct a maximal counterpart to the minimal quantum group-twisted
tensor product of $C^{*}$-algebras studied by Meyer, Roy and Woronowicz, which
is universal with respect to representations satisfying braided commutation
relations. Much like the minimal one, this product yields a monoidal structure
on the coactions of a quasi-triangular $C^{*}$-quantum group, the horizontal
composition in a bicategory of Yetter-Drinfeld $C^{*}$-algebras, and coincides
with a Rieffel deformation of the non-twisted tensor product in the case of
group coactions.
