  We study the real-time simulation of open quantum systems, where the system
is modeled by a spin chain, with each spin associated with its own harmonic
bath. Our method couples the inchworm method for the spin-boson model and the
modular path integral methodology for spin systems. In particular, the
introduction of the inchworm method can significantly suppress the numerical
sign problem. Both methods are tweaked to make them work seamlessly with each
other. We represent our approach in the language of diagrammatic methods, and
analyze the asymptotic behavior of the computational cost. Extensive numerical
experiments are done to validate our method.
