  The doubled target space of the fundamental closed string is identified with
its phase space and described by an almost para-Hermitian geometry. We explore
this setup in the context of group manifolds which admit a maximally isotropic
subgroup. This leads to a formulation of the Poisson-Lie $\sigma$-model and
Poisson-Lie T-duality in terms of para-Hermitian geometry. The emphasis is put
on so called half-integrable setups where only one of the Lagrangian subspaces
of the doubled space has to be integrable. Using the dressing coset
construction in Poisson-Lie T-duality, we extend our construction to more
general coset spaces. This allows to explicitly obtain a huge class of
para-Hermitian geometries. Each of them is automatically equipped which a
generalized frame field, required for consistent generalized Scherk-Schwarz
reductions. As examples we present integrable $\lambda$- and
$\eta$-deformations on the three- and two-sphere.
