It is expected that taut foliations on 3-manifolds typically admit (almost) transverse pseudo-Anosov flows. This motivates a natural question: given a pseudo-Anosov flow $\phi$ on a 3-manifold $M$ and a suitable link $L \subset M$ of closed orbits of $\phi$, for which Dehn surgery multislopes along $L$ does the surgered manifold admit a taut foliation transverse to the induced flow? The set of such multislopes has a remarkable staircase structure with corners at rational multislopes that accumulate at very specific points. These sets can be algorithmically computed in many small examples. In work in preparation with Jonathan Zung, we explain some key features of these 'ziggurat' sets and justify their name, using tools from contact geometry.