A folklore conjecture of Gabai--Mosher asserts that taut foliations on $3$-manifolds typically admit (almost) transverse pseudo-Anosov flows. It is natural to ask: given a pseudo-Anosov flow $\phi$ on a $3$-manifold $M$ and a suitable link $L \subset M$ of closed orbits of $\phi$, which Dehn surgery multislopes along $L$ admit taut foliations transverse to the flow? This set of multislopes has a remarkable staircase structure whose corners are rational multislopes which accumulate at very specific points. It can also be algorithmically computed in many small examples. In work in preparation with Jonathan Zung, we explain some key features of these sets and justify their name of ziggurats using tools from contact geometry.