  Consider the full primitive equations, i.e. the three dimensional primitive
equations coupled to the equation for temperature and salinity, and subject to
outer forces. It is shown that this set of equations is globally strongly
well-posed for arbitrary large initial data lying in certain interpolation
spaces, which are explicitly characterized as subspaces of $H^{2/p,p}$,
$1<p<\infty$, satisfying certain boundary conditions. In particular, global
well-posedeness of the full primitive equations is obtained for initial data
having less differentiability properties than $H^1$, hereby generalizing by
result by Cao and Titi (Ann. of Math. (2) 166 (2007), no. 1, 245-267) to the
case of non-smooth data. In addition, it is shown that the solutions are
exponentially decaying provided the outer forces possess this property.
