  We engineer compact contours on the moduli spaces of genus-zero Riemann
surfaces that achieve analytic continuation from Euclidean to Lorentzian
worldsheets. These generalized Pochhammer contours are based on the
combinatorics of associahedra and make the analytic properties of tree-level
amplitudes entirely manifest for any number and type of external strings. We
use them in practice to perform first numerical computations of open and closed
string amplitudes directly in the physical kinematics for $n=4,5,6,7,8,9$. We
provide a code that allows anyone to do such computations.
