Efficiently deciding if an ideal is toric after a linear coordinate change

We propose an effective algorithm that decides if a prime ideal in a polynomial ring over the complex numbers can be transformed into a toric ideal by a linear automorphism of the ambient space. If this is the case, the algorithm computes such a transformation explicitly. The algorithm can compute that all Gaussian graphical models on five vertices that are not initially toric cannot be made toric by any linear coordinate change. The same holds for all Gaussian conditional independence ideals of undirected graphs on six vertices.