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

We construct a master dynamical system on a $\operatorname{U}(n)$ quasi-Poisson manifold, $\mathcal{M}_d$, built from the double $\operatorname{U}(n) \times \operatorname{U}(n)$ and $d\geq 2$ open balls in $\mathbb{C}^n$, whose quasi-Poisson structures are obtained from $T^* \mathbb{R}^n$ by exponentiation. A pencil of quasi-Poisson bivectors $P_{\underline{z}}$ is defined on $\mathcal{M}_d$ that depends on $d(d-1)/2$ arbitrary real parameters and gives rise to pairwise compatible Poisson brackets on the $\operatorname{U}(n)$-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 $T^*\!\operatorname{U}(n) \times \mathbb{C}^{n\times d}$. Its commuting Hamiltonians are pullbacks of the class functions on one of the $\operatorname{U}(n)$ 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 $\mathcal{M}_d/\operatorname{U}(n)$ 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 $P_{\underline{z}}$. The restrictions of the reduced system on minimal symplectic leaves parameterized by generic elements of the center of $\operatorname{U}(n)$ 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.