  It was established by Boalch that Euler continuants arise as Lie group valued
moment maps for a class of wild character varieties described as moduli spaces
of points on $\mathbb{P}^1$ by Sibuya. Furthermore, Boalch noticed that these
varieties are multiplicative analogues of certain Nakajima quiver varieties
originally introduced by Calabi, which are attached to the quiver $\Gamma_n$ on
two vertices and $n$ equioriented arrows. In this article, we go a step further
by unveiling that the Sibuya varieties can be understood using noncommutative
quasi-Poisson geometry modeled on the quiver $\Gamma_n$. We prove that the
Poisson structure carried by these varieties is induced, via the
Kontsevich-Rosenberg principle, by an explicit Hamiltonian double quasi-Poisson
algebra defined at the level of the quiver $\Gamma_n$ such that its
noncommutative multiplicative moment map is given in terms of Euler
continuants. This result generalises the Hamiltonian double quasi-Poisson
algebra associated with the quiver $\Gamma_1$ by Van den Bergh. Moreover, using
the method of fusion, we prove that the Hamiltonian double quasi-Poisson
algebra attached to $\Gamma_n$ admits a factorisation in terms of $n$ copies of
the algebra attached to $\Gamma_1$.
