  We derive the torsion constraints and show the consistency of equations of
motion of four-dimensional Type II supergravity in superspace, with Type II
sigma model. This is achieved by coupling the four-dimensional compactified
Type II Berkovits' superstring to an N=2 curved background and requiring that
the sigma-model has superconformal invariance at tree-level. We compute this in
a manifestly 4D N=2 supersymmetric way. The constraints break the target
conformal and SU(2) invariances and the dilaton will be a conformal,
$SU(2)\times U(1)$ compensator. For Type II superstring in four dimensions,
worldsheet supersymmetry requires two different compensators. One type is
described by chiral and anti-chiral superfields. This compensator can be
identified with a vector multiplet. The other Type II compensator is described
by twist-chiral and twist-anti-chiral superfields and can be identified with a
tensor hypermultiplet. Also, the superconformal invariance at tree-level
selects a particular gauge, where the matter is fixed, but not the
compensators. After imposing the reality conditions, we show that the Type II
sigma model at tree-level is consistent with the equations of motion for Type
II supergravity in the string gauge.
