  Given a standard complex semisimple Poisson Lie group $(G, \pi_{st})$,
generalised double Bruhat cells $G^{u, v}$ and generalised Bruhat cells $O^u$
equipped with naturally defined holomorphic Poisson structures, where u, v are
finite sequences of Weyl group elements, were defined and studied by Jiang Hua
Lu and the author. We prove in this paper that $G^{u,u}$ is naturally a Poisson
groupoid over $O^u$, extending a result from the aforementioned authors about
double Bruhat cells in $(G, \pi_{st})$. Our result on $G^{u,u}$ is obtained as
an application of a construction interesting in its own right, of a local
Poisson groupoid over a mixed product Poisson structure associated to the
action of a pair of Lie bialgebras. This construction involves using a local
Lagrangian bisection in a double symplectic groupoid closely related to the
global R-matrix studied by Weinstein and Xu, to twist a direct product of
Poisson groupoids.
