Real and complex rank for real symmetric tensors with low ranks

We study the case of a real homogeneous polynomial $P$ whose minimal real and complex decompositions in terms of powers of linear forms are different. We prove that, if the sum of the complex and the real ranks of $P$ is at most $3\mathrm{deg}(P)-1$, then the difference of the two decompositions is completely determined either on a line or on a conic or two disjoint lines.