  We establish the existence and uniqueness of pathwise strong solutions to the
stochastic 3D primitive equations with only horizontal viscosity and
diffusivity driven by transport noise on a cylindrical domain $M=(-h,0) \times
G$, $G\subset \mathbb{R}^2$ bounded and smooth, with the physical Dirichlet
boundary conditions on the lateral part of the boundary. Compared to the
deterministic case where the uniqueness of $z$-weak solutions holds in $L^2$,
more regular initial data are necessary to establish uniqueness in the
anisotropic space $H^1_z L^2_{xy}$ so that the existence of local pathwise
solutions can be deduced from the Gy\"{o}ngy-Krylov theorem. Global existence
is established using the logarithmic Sobolev embedding, the stochastic Gronwall
lemma and an iterated stopping time argument.
