Trigonometric real form of the spin RS model of Krichever and Zabrodin

We investigate the trigonometric real form of the spin Ruijsenaars-Schneider system introduced, at the level of equations of motion, by Krichever and Zabrodin in 1995. This pioneering work and all earlier studies of the Hamiltonian interpretation of the system were performed in complex holomorphic settings; understanding the real forms is a non-trivial problem. We explain that the trigonometric real form emerges from Hamiltonian reduction of an obviously integrable 'free' system carried by a spin extension of the Heisenberg double of the ${\rm U}(n)$ Poisson-Lie group. The Poisson structure on the unreduced real phase space ${\rm GL}(n,\mathbb{C}) \times \mathbb{C}^{nd}$ is the direct product of that of the Heisenberg double and $d\geq 2$ copies of a ${\rm U}(n)$ covariant Poisson structure on $\mathbb{C}^n \simeq \mathbb{R}^{2n}$ found by Zakrzewski, also in 1995. We reduce by fixing a group valued moment map to a multiple of the identity, and analyze the resulting reduced system in detail. In particular, we derive on the reduced phase space the Hamiltonian structure of the trigonometric spin Ruijsenaars-Schneider system and we prove its degenerate integrability.