The use of the relativistic and spin-polarized real-space Korringa-Kohn-Rostoker (KKR) method is limited to small systems (less than 100 atoms). This is due to the prohibitively large CPU times needed for the inversion of the KKR matrix. To study systems of more than a thousand atoms, we have implemented the concept of a screened reference medium, within the fully relativistic spin-polarized version of the real-space locally self-consistent multiple-scattering method (LSMS). The LSMS method makes use of a local interaction zone (LIZ) for solving the quantum mechanical problem, while the Poisson equation is solved in the whole space. The screened reference medium gives rise to sparse KKR matrices and using state-of-the-art sparse matrix technology a substantial reduction in the CPU times is obtained, enabling applications of the method to systems whose LIZ consists of more than a thousand atoms. The method is benchmarked by application to the elemental transition metals, the fcc (face-centred-cubic) Co and Ni, and the bcc (body-centred cubic) Fe, and compared to the results of the conventional k-space methods. The convergence in real space of the magnetic moments, the magnetocrystalline anisotropy energy and the orbital moment anisotropy is discussed in detail.