  A geometric interpretation of the duality between two real forms of the
complex trigonometric Ruijsenaars-Schneider system is presented. The phase
spaces of the systems in duality are viewed as two different models of the same
reduced phase space arising from a suitable symplectic reduction of the
standard Heisenberg double of U(n). The collections of commuting Hamiltonians
of the systems in duality are shown to descend from two families of `free'
Hamiltonians on the double which are dual to each other in a Poisson-Lie sense.
Our results give rise to a major simplification of Ruijsenaars' proof of the
crucial symplectomorphism property of the duality map.
