  Previous studies of the type IIB superstring in an ${AdS_5 \times S^5}$
background are based on a description of the superspace geometry as the
quotient space $PSU(2,2|4)/SO(4,1) \times SO(5)$. This paper develops an
alternative approach in which the Grassmann coordinates provide a nonlinear
realization of $PSU(2,2|4)$ based on the quotient space $PSU(2,2|4)/SU(2,2)
\times SU(4)$, and the bosonic coordinates are described as a submanifold of
$SU(2,2) \times SU(4)$. This formulation keeps all bosonic symmetries manifest,
and it provides the complete dependence on the Grassmann coordinates in terms
of simple analytic expressions. It is used to construct the superstring
world-sheet action in a form in which the $PSU(2,2|4)$ symmetry is manifest and
kappa symmetry can be established. This formulation might have some advantages
compared to previous ones, but this remains to be demonstrated.
