  Capturing non-Markovian dynamics of open quantum systems is generally a
challenging problem, especially for strongly-interacting many-body systems. In
this work, we combine recently developed non-Markovian quantum state diffusion
techniques with tensor network methods to address this challenge. As a first
example, we explore a Hubbard-Holstein model with dissipative phonon modes,
where this new approach allows us to quantitatively assess how correlations
spread in the presence of non-Markovian dissipation in a 1D many-body system.
We find regimes where correlation growth can be enhanced by these effects,
offering new routes for dissipatively enhancing transport and correlation
spreading, relevant for both solid state and cold atom experiments.
