Taut foliations, left-orders, and pseudo-Anosov mapping tori

For a large class of 3-manifolds with taut foliations, we construct an action of ${\pi }_1(M)$ on $\mathbb{R}$ by orientation preserving homeomorphisms which captures the transverse geometry of the leaves. This action is complementary to Thurston’s universal circle. Applications include the left-orderability of the fundamental groups of every non-trivial surgery on the figure eight knot. Our techniques also apply to at least 2598 manifolds representing 44.7% of the non-L-space rational homology spheres in the Hodgson-Weeks census of small closed hyperbolic 3-manifolds.