  We consider non(anti)commutative superspace with coordinate dependent
deformation parameters $C^{\alpha\beta}$. We show that a chiral ${\cal N}=1/2$
supersymmetry can be defined and that chiral and antichiral superfields are
still closed under the Moyal-Weyl associative product implementing the
deformation. A consistent ${\cal N}=1/2$ Super Yang-Mills deformed theory can
be constructed provided $C^{\alpha\beta}$ satisfies a suitable condition which
can be connected with the graviphoton background at the origin of the
deformation. After adding matter we also discuss the Konishi anomaly and the
gluino condensation.
