In this paper, a new variant of the Jacobi-Davidson (JD) method is presented that is specifically designed for real unsymmetric matrix pencils. Whenever a pencil has a complex conjugate pair of eigenvalues, the method computes the two-dimensional real invariant subspace spanned by the two corresponding complex conjugated eigenvectors. This is beneficial for memory costs and in many cases it also accelerates the convergence of the JD method. Both real and complex formulations of the correction equation are considered. In numerical experiments, the RJDQZ variant is compared with the original JDQZ method.