  The spin-boson model, describing a two-level system strongly coupled to a
bosonic bath, is extensively studied as a paradigmatic dissipative quantum
system, exhibiting rich dynamical behavior and even a localization transition
in the strong coupling regime. Here, we additionally consider dephasing as a
source of Markovian dissipation on top of the non-Markovian dynamics due to an
Ohmic bath, and investigate the dynamics of the spin. We show that the
characteristic frequency of the spin dynamics, while strongly renormalized by
the bosonic bath, changes in a simple fashion (or doesn't change at all) with
dephasing. To obtain these results, we develop an exact non-perturbative method
known as the stochastic Schr\"{o}dinger equation, mimicking the Ohmic bath via
a stochastic magnetic field combined with the Lindblad quantum master equation
due to dephasing, which allows us to numerically compute the dynamics.
Furthermore, we derive weak-coupling analytic results utilizing the well-known
non-interacting blip approximation. Our findings are relevant to quantum
simulation of the spin-boson model in the regime of strong coupling in trapped
ions and circuit QED architectures among others.
