  We construct explicit combinatorial Bernoulli factories for the class of
\emph{flow-based polytopes}; integral 0/1-polytopes defined by a set of network
flow constraints. This generalizes the results of Niazadeh et al. (who
constructed an explicit factory for the specific case of bipartite perfect
matchings) and provides novel exact sampling procedures for sampling paths,
circulations, and $k$-flows. In the process, we uncover new connections to
algebraic combinatorics.
