Planet formation is often considered in the context of one circumstellar disc around one star. Yet, stellar binary systems are ubiquitous, and thus a substantial fraction of all potential planets must form and evolve in more complex, dynamical environments. We present the results of a 5 yr astrometric monitoring campaign studying 45 binary star systems that host Kepler planet candidates. The planet-forming environments in these systems would have literally been shaped by the binary orbits that persist to the present day. Crucially, the mutual inclinations of star-planet orbits can only be addressed by a statistical sample. We describe in detail our sample selection and Keck/NIRC2 laser guide star adaptive optics observations collected from 2012 to 2017. We measure orbital arcs, with a typical accuracy of ∼0.1 mas yr-1, that test whether the binary orbits tend to be aligned with the edge-on transiting planet orbits. We rule out randomly distributed binary orbits at 4.7σ, and we show that low mutual inclinations are required to explain the observed orbital arcs. If the stellar orbits have a field binary-like eccentricity distribution, then the best match to our observed orbital arcs is a distribution of mutual inclinations ranging from 0° to 30°. We discuss the implications of such widespread planet-binary alignment in the theoretical context of planet formation and circumstellar disc evolution
