  We introduce a bi-Hamiltonian hierarchy on the cotangent bundle of the real
Lie group ${\mathrm{GL}}(n,{\mathbb{C}})$, and study its Poisson reduction with
respect to the action of the product group ${{\mathrm U}(n)} \times {{\mathrm
U}(n)}$ arising from left- and right-multiplications. One of the pertinent
Poisson structures is the canonical one, while the other is suitably
transferred from the real Heisenberg double of ${\mathrm{GL}}(n,{\mathbb{C}})$.
When taking the quotient of $T^*{\mathrm{GL}}(n,{\mathbb{C}})$ we focus on the
dense open subset of ${\mathrm{GL}}(n,{\mathbb{C}})$ whose elements have
pairwise distinct singular values. We develop a convenient description of the
Poisson algebras of the ${{\mathrm U}(n)} \times {{\mathrm U}(n)}$ invariant
functions, and show that one of the Hamiltonians of the reduced bi-Hamiltonian
hierarchy yields a hyperbolic Sutherland model coupled to two ${\mathfrak
u}(n)^*$-valued spins. Thus we obtain a new bi-Hamiltonian interpretation of
this model, which represents a special case of Sutherland models coupled to two
spins obtained earlier from reductions of cotangent bundles of reductive Lie
groups equipped with their canonical Poisson structure. Upon setting one of the
spins to zero, we recover the bi-Hamiltonian structure of the standard
hyperbolic spin Sutherland model that was derived recently by a different
method.
