  We show that a certain superfield formalism can be used to find an off-shell
supersymmetric description for some supersymmetric field theories where
conventional superfield formalism does not work. This "new" formalism contains
even auxiliary variables in addition to conventional odd super-coordinates. The
idea of this construction is similar to the pure spinor formalism developed by
N.Berkovits. It is demonstrated that using this formalism it is possible to
prove that the certain Chern-Simons-like (Witten's OSFT-like) theory can be
considered as an off-shell version for some on-shell supersymmetric field
theories. We use the simplest non-trivial model found in [2] to illustrate the
power of this pure spinor superfield formalism. Then we redo all the
calculations for the case of 10-dimensional Super-Yang-Mills theory. The
construction of off-shell description for this theory is more subtle in
comparison with the model of [2] and requires additional Z_2 projection. We
discover experimentally (through a direct explicit calculation) a non-trivial
Z_2 duality at the level of Feynman diagrams. The nature of this duality
requires a better investigation.
