  In this paper we establish global well-posedness and instantaneous
regularization results for the primitive equations with transport noise of
H\"{o}lder regularity $ \gamma>\frac{1}{2}$. It is known that if $\gamma<1$,
then the noise is too rough for a strong formulation of primitive equations in
an $L^2$-based setting. To handle rough noise, we crucially use
$L^q$-techniques with $q> 2$. Interestingly, we identify a family of critical
anisotropic Besov spaces for primitive equations, which is new even in the
deterministic case. The behavior of these spaces reflects the intrinsic
anisotropy of the primitive equations and plays an essential role in
establishing global well-posedness and regularization. Our results cover
Kraichnan's type noise with correlation greater than one, and as a by-product,
a 2D noise reproducing the Kolmogorov spectrum of turbulence. Moreover, the
instantaneous regularization is new also in the widely studied case of
$H^1$-data and $\gamma>1 $.
