  The real-time propagator of the symmetric Rosen-Morse, also known as the
symmetric modified P\"oschl-Teller, barrier is expressed in the
Picard-Lefschetz path integral formalism using real and complex classical
paths. We explain how the interference pattern in the real-time propagator and
energy propagator is organized by caustics and Stoke's phenomena, and list the
relevant real and complex classical paths as a function of the initial and
final position. We discover the occurrence of singularity crossings, where the
analytic continuation of the complex classical path no longer satisfies the
boundary value problem and needs to be analytically continued. Moreover, we
demonstrate how these singularity crossings play a central role in the
real-time description of quantum tunneling.
