  The toric fiber product is a general procedure for gluing two ideals,
homogeneous with respect to the same multigrading, to produce a new homogeneous
ideal. Toric fiber products generalize familiar constructions in commutative
algebra like adding monomial ideals and the Segre product. We describe how to
obtain generating sets of toric fiber products in non-zero codimension and
discuss persistence of normality and primary decompositions under toric fiber
products.
  Several applications are discussed, including (a) the construction of Markov
bases of hierarchical models in many new cases, (b) a new proof of the quartic
generation of binary graph models associated to $K_{4}$-minor free graphs, and
(c) the recursive computation of primary decompositions of conditional
independence ideals.
