We construct a master dynamical system on a U(n) quasi-Poisson manifold, Md, built from the double U(n) x U(n) and d >= 2 open balls in Cn, whose quasi-Poisson structures are obtained from T *R-n by exponentiation. A pencil of quasi-Poisson bivectors P-z is defined on M d that depends on d(d - 1)/2 arbitrary real parameters and gives rise to pairwise compatible Poisson brackets on the U(n)-invariant functions. The master system on M-d is a quasi-Poisson analogue of the degenerate integrable system of free motion on the extended cotangent bundle T * U(n) x C-nxd. Its commuting Hamiltonians are pullbacks of the class functions on one of the 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 M d/ 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-z. The restrictions of the reduced system on minimal symplectic leaves parameterized by generic elements of the center of 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.