  We extend the scope of full configuration interaction quantum Monte Carlo
(FCIQMC) to be applied to coupled fermion-boson hamiltonians, alleviating the a
priori truncation in boson occupation which is necessary for many other wave
function based approaches to be tractable. Detailing the required algorithmic
changes for efficient excitation generation, we apply FCIQMC in two contrasting
settings. The first is a sign-problem-free Hubbard--Holstein model of local
electron-phonon interactions, where we show that with care to control for
population bias via importance sampling and/or reweighting, the method can
achieve unbiased energies extrapolated to the thermodynamic limit, without
suffering additional computational overheads from relaxing boson occupation
constraints. Secondly, we apply the method as a `solver' within a quantum
embedding scheme which maps electronic systems to local electron-boson
auxiliary models, with the bosons representing coupling to long-range
plasmonic-like fluctuations. We are able to sample these general electron-boson
hamiltonians with ease despite a formal sign problem, including a faithful
reconstruction of converged reduced density matrices of the system.
