The presence of monodromy means that the definition of global contact orders (those specified globally for the whole moduli space) with a reduced subscheme \(Z\subset X\) as global sections of \(\Gamma(Z, (f^*\overline{\mathcal M}_X)|_Z)\) no longer works apparently. For a toric variety target, the contact orders should be vectors in the cocharacter lattice, so the fix here is to simply take a colimit and thereby identify all the vectors related by monodromy.