  We study the $N=2$ string with a real structure on the $(2,2)$ spacetime,
using BRST methods. Several new features emerge. In the diagonal basis, the
operator $\exp(\lambda \int^z J^{\rm tot})$, which is associated with the
moduli for the $U(1)$ gauge field on the world-sheet, is local and it relates
the physical operators in the NS and R sectors. However, the picture-changing
operators are non-invertible in this case, and physical operators in different
pictures cannot be identified. The three-point interactions of all physical
operators leads to three different types of amplitudes, which can be
effectively described by the interactions of a scalar NS operator and a bosonic
spinorial R operator. In the off-diagonal bases for the fermionic currents, the
picture-changing operators are invertible, and hence physical operators in
different pictures can be identified. However, now there is no local operator
$\exp(\lambda \int^z J^{\rm tot})$ that relates the physical operators in
different sectors. The physical spectrum is thus described by one scalar NS
operator and one spinorial R operator. The NS and R operators give rise to
different types of three-point amplitudes, and thus cannot be identified.
