  To describe the double shuffle relations between multiple polylogarithm
values at $N$th roots of unity, Racinet attached to each finite cyclic group
$G$ of order $N$ and each group embedding $\iota : G \to \mathbb{C}^{\times}$,
a $\mathbb{Q}$-scheme $\mathsf{DMR}^{\iota}$ which associates to each
commutative $\mathbb{Q}$-algebra $\mathbf{k}$, a set
$\mathsf{DMR}^{\iota}(\mathbf{k})$ that can be decomposed as a disjoint union
of sets $\mathsf{DMR}^{\iota}_{\lambda}(\mathbf{k})$ with $\lambda \in
\mathbf{k}$. He also exhibited a $\mathbb{Q}$-group scheme $\mathsf{DMR}_0^G$
and showed that $\mathsf{DMR}^{\iota}_{\lambda}(\mathbf{k})$ is a torsor for
the action of $\mathsf{DMR}_0^G(\mathbf{k})$. Then, Enriquez and Furusho showed
for $N=1$ that a subscheme $\mathsf{DMR}^{\iota}_{\times}$ of
$\mathsf{DMR}^{\iota}$ is a torsor of isomorphisms relating de Rham and Betti
objects. In previous work, we reformulated Racinet's construction in terms of
crossed products and identified his coproduct with a coproduct
$\widehat{\Delta}^{\mathcal{M}, \mathrm{DR}}_G$ defined on a module
$\widehat{\mathcal{M}}_G^{\mathrm{DR}}$ over an algebra
$\widehat{\mathcal{W}}_G^{\mathrm{DR}}$ equipped with its own coproduct
$\widehat{\Delta}^{\mathcal{W}, \mathrm{DR}}_G$. In this paper, we provide a
generalization of Enriquez and Furusho's result to any $N \geq 1$: we exhibit a
module $\widehat{\mathcal{M}}_N^{\mathrm{B}}$ over an algebra
$\widehat{\mathcal{W}}_N^{\mathrm{B}}$ and show the existence of compatible
coproducts $\widehat{\Delta}^{\mathcal{W}, \mathrm{B}}_N$ and
$\widehat{\Delta}^{\mathcal{M}, \mathrm{B}}_N$ such that
$\mathsf{DMR}^{\iota}_{\times}$ is contained in the torsor of isomorphisms
relating $\widehat{\Delta}^{\mathcal{W}, \mathrm{B}}_N$ (resp.
$\widehat{\Delta}^{\mathcal{M}, \mathrm{B}}_N$) to
$\widehat{\Delta}^{\mathcal{W}, \mathrm{DR}}_G$ (resp.
$\widehat{\Delta}^{\mathcal{M}, \mathrm{DR}}_G$).
