  Integrable many-body systems of Ruijsenaars--Schneider--van Diejen type
displaying action-angle duality are derived by Hamiltonian reduction of the
Heisenberg double of the Poisson-Lie group $\mathrm{SU}(2n)$. New global models
of the reduced phase space are described, revealing non-trivial features of the
two systems in duality with one another. For example, after establishing that
the symplectic vector space $\mathbb{C}^n\simeq\mathbb{R}^{2n}$ underlies both
global models, it is seen that for both systems the action variables generate
the standard torus action on $\mathbb{C}^n$, and the fixed point of this action
corresponds to the unique equilibrium positions of the pertinent systems. The
systems in duality are found to be non-degenerate in the sense that the
functional dimension of the Poisson algebra of their conserved quantities is
equal to half the dimension of the phase space. The dual of the deformed
Sutherland system is shown to be a limiting case of a van Diejen system.
