  Characteristic imsets are 0-1 vectors which correspond to Markov equivalence
classes of directed acyclic graphs. The study of their convex hull, named the
characteristic imset polytope, has led to new and interesting geometric
perspectives on the important problem of causal discovery. In this paper we
begin the study of the associated toric ideal. We develop a new generalization
of the toric fiber product, which we call a quasi-independence gluing, and show
that under certain combinatorial homogeneity conditions, one can iteratively
compute a Gr\"obner basis via lifting. For faces of the characteristic imset
polytope associated to trees, we apply this technique to compute a Gr\"obner
basis for the associated toric ideal. We end with a study of the characteristic
ideal of the cycle and propose directions for future work.
