Irreducible train tracks for pseudo-Anosov homeomorphisms

We describe a construction of invariant train tracks with irreducible transition matrix for pseudo-Anosov homeomorphisms. This fills what seems to be a gap in the literature concerning the existence of such train tracks. The construction starts with an invariant train track associated to the veering triangulation of the mapping torus of the homeomorphism and then uses the veering property to characterize branches which obstruct irreducibility, finally modifying the track to bypass these obstructions.