  The bi-Yang-Baxter sigma-model is a certain two-parameter deformation of the
principal chiral model on a real Lie group G for which the left and right
G-symmetries of the latter are both replaced by Poisson-Lie symmetries. It was
introduced by C. Klimcik who also recently showed it admits a Lax pair, thereby
proving it is integrable at the Lagrangian level. By working in the Hamiltonian
formalism and starting from an equivalent description of the model as a
two-parameter deformation of the coset sigma-model on G x G / G_diag, we show
that it also admits a Lax matrix whose Poisson bracket is of the standard
r/s-form characterised by a twist function which we determine. A number of
results immediately follow from this, including the identification of certain
complex Poisson commuting Kac-Moody currents as well as an explicit description
of the q-deformed symmetries of the model. Moreover, the model is also shown to
fit naturally in the general scheme recently developed for constructing
integrable deformations of sigma-models. Finally, we show that although the
Poisson bracket of the Lax matrix still takes the r/s-form after fixing the
G_diag gauge symmetry, it is no longer characterised by a twist function.
