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 Gröbner basis
via lifting. For faces of the characteristic imset polytope associated to trees, we apply
this technique to compute a Gröbner basis for the associated toric ideal. We end
with a study of the characteristic ideal of the cycle and provide direction for future
work.