  We present a theory of N=2 chiral supergravity in (10+2)-dimensions. This
formulation is similar to N=1 supergravity presented recently using
null-vectors in 12D. In order to see the consistency of this theory, we perform
a simple dimensional reduction to ten-dimensions, reproducing the type IIB
chiral supergravity. We also show that our supergravity can be consistent
background for super (2+2)-brane theory, satisfying fermionic invariance of the
total action. Such supergravity theory without manifest Lorentz invariance had
been predicted by the recent F-theory in twelve-dimensions.
