  There is much discussion of scenarios where the space-time coordinates x^\mu
are noncommutative. The discussion has been extended to include nontrivial
anticommutation relations among spinor coordinates in superspace. A number of
authors have studied field theoretical consequences of the deformation of N=1
superspace arising from nonanticommutativity of coordinates \theta, while
leaving \bar{theta}'s anticommuting. This is possible in Euclidean superspace
only. In this note we present a way to extend the discussion by making both
\theta and \bar{theta} coordinates non-anticommuting in Minkowski superspace.
We present a consistent algebra for the supercoordinates, find a star-product,
and give the Wess-Zumino Lagrangian L_{WZ} within our model. It has two extra
terms due to non(anti)commutativity. The Lagrangian in Minkowski superspace is
always manifestly Hermitian and for L_{WZ} it preserves Lorentz invariance.
