A matrix is called totally nonnegative if all the determinants of its square submatrices are nonnegative. One can show, using the celebrated Lindström-Gessel-Viennot lemma, that any invertible totally nonnegative matrix can be realized as the path matrix of a planar, acyclic digraph with positive weights. A result by Brenti broadens this statement to include non-invertible, non-square matrices. Lam and Pylyavskyy later introduced a notion of total positivity for matrices with real polynomial entries, and showed that when they are invertible, they are representable by cylindrical (rather than planar) digraphs. We expand their result to include non-square, non-invertible matrices as well. The proof involves certain combinatorially-defined matrix functions called Temperley-Lieb immanants.