Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $\mathrm{U}(n)$

We construct a master dynamical system on a $\mathrm{U}(n)$ quasi-Poisson manifold, ${\mathcal{M}}_d$, built from the double $\mathrm{U}(n)\times \mathrm{U}(n)$ and $d\ge 2$ open balls in ${\mathbb{C}}^n$, whose quasi-Poisson structures are obtained from $T^{\ast }{\mathbb{R}}^n$ by exponentiation. A pencil of quasi-Poisson bivectors $P_{\underset{‾}{z}}$ is defined on ${\mathcal{M}}_d$ that depends on $d(d-1)\mathrm{/}2$ arbitrary real parameters and gives rise to pairwise compatible Poisson brackets on the $\mathrm{U}(n)$-invariant functions. The master system on ${\mathcal{M}}_d$ is a quasi-Poisson analogue of the degenerate integrable system of free motion on the extended cotangent bundle $T^{\ast }\text{\hspace{0.17em}⁣}\mathrm{U}(n)\times {\mathbb{C}}^{n\times d}$. Its commuting Hamiltonians are pullbacks of the class functions on one of the $\mathrm{U}(n)$ factors. We prove that the master system descends to a degenerate integrable system on a dense open subset of the smooth component of the quotient space ${\mathcal{M}}_d\mathrm{/}\mathrm{U}(n)$ associated with the principal orbit type. Any reduced Hamiltonian arising from a class function generates the same flow via any of the compatible Poisson structures stemming from the bivectors $P_{\underset{‾}{z}}$. The restrictions of the reduced system on minimal symplectic leaves parameterized by generic elements of the center of $\mathrm{U}(n)$ provide a new real form of the complex, trigonometric spin Ruijsenaars-Schneider model of Krichever and Zabrodin. This generalizes the derivation of the compactified trigonometric RS model found previously in the $d=1$ case.