In this article we classify all differential real cubic systems possessing two affine real non-parallel invariant straight lines of maximal multiplicity. We show that the maximal multiplicity of each of these lines is at most three. The maximal sequences of multiplicities: m(3, 3; 1), m(3, 2; 2), m(3, 1; 3), m(2, 2; 3), m1(2, 1; 3), m1(1, 1; 3) are determined. The normal forms and the corresponding perturbations of the cubic systems which realize these cases are given.