  We construct supergravity theories in twelve and thirteen dimensions with the
respective signatures (10,2) and (11,2) with some technical details. Starting
with N=1 supergravity in 10+2 dimensions coupled to Green-Schwarz superstring,
we give N=2 chiral supergravity in 10+2 dimensions with its couplings to super
(2+2)-brane. We also build an N=1 supergravity in 11+2 dimensions, coupled to
supermembrane. All of these formulations utilize scalar (super)fields intact
under supersymmetry, replacing the null-vectors introduced in their original
formulations. This method makes all the equations SO(10,2) or SO(11,2) Lorentz
covariant, up to modified Lorentz generators. We inspect the internal
consistency of these formulations, in particular with the usage of the modified
Lorentz generators for the extra coordinates.
