A version of the Penrose transform is introduced in split signature. It relates cohomological data on CP3 \ RP3 and the kernel of differential operators on M, the (real) Grassmannian of 2-planes in R-4. As an example we derive the following cohomological interpretation of the so-called X-ray transform
H-c(1)(CP3\RP3, O(-2)) ->congruent to ker(rectangle(2.2) : Gamma(omega) (M, (epsilon|-1|) over tilde) -> Gamma(omega) (M, (epsilon|-3|) over tilde))
where Gamma(omega) (M, (epsilon|-1|) over tilde) and Gamma(omega) (M, (epsilon|-3|) over tilde) are real analytic sections of certain (homogeneous) line bundles on M, c stands for cohomology with compact support and rectangle(2.2) is the ultrahyperbolic operator. Furthermore, this gives a cohomological realization of the so-called "minimal" representation of SL(4, R). We also present the split Penrose transform in split instanton backgrounds.