  Let $G$ be a simple complex Lie group, $\alg{g}$ be its Lie algebra, $K$ be a
maximal compact form of $G$ and $\alg{k}$ be a Lie algebra of $K$. We denote by
$X\rightarrow \overline{X}$ the anti-involution of $\alg{g}$ which singles out
the compact form $\alg{k}$. Consider the space of flat $\alg{g}$-valued
connections on a Riemann sphere with three holes which satisfy the additional
condition $\overline{A(z)}=-A(\overline{z})$. We call the quotient of this
space over the action of the gauge group $\overline{g(z)}=g^{-1}(\overline{z})$
a \emph{hyperbolic} moduli space of flat connections. We prove that the
following three symplectic spaces are isomorphic: 1. The hyperbolic moduli
space of flat connections. 2. The symplectic multiplicity space obtained as
symplectic quotient of the triple product of co-adjoint orbits of $K$. 3. The
Poisson-Lie multiplicity space equal to the Poisson quotient of the triple
product of dressing orbits of $K$.
