Veering triangulations and transverse foliations

We present a combinatorial approach to the existence of foliations and contact structures transverse to a given pseudo-Anosov flow. Let $\varphi$ be a transitive pseudo-Anosov flow on a closed oriented 3-manifold. Our main technical result is that every codimension 1 foliation transverse to $\varphi$ is carried by a single branched surface coming from a veering triangulation. Combined with recent breakthrough work of Massoni, this reduces the existence problem for transverse foliations to something like the feasibility of a system of inequalities (rather than equations!) over $Homeo_+([0,1])$. As a proof of concept, we show that for the hyperbolic, fibered, non-L-space knot $10_{145}$, the natural pseudo-Anosov flow on the slope $s$ Dehn surgery admits a transverse foliation for $s\in (-\infty, 3)$, but does not admit such a foliation for $s\in [5,\infty)$. The negative result is part of a more general Milnor--Wood type phenomenon which puts limitations on some well known methods for constructing taut foliations on Dehn surgeries.