  Obtaining dynamics of an interacting quantum many-body system connected to
multiple baths initially at different, finite, temperatures and chemical
potentials is a challenging problem. This is due to a combination of the
prevalence of strong correlations in the system, the infinite nature of the
baths and the long time to reach steady state. In this work we develop a
general formalism that allows access to the full non-Markovian dynamics of such
open quantum many-body systems up to the non-equilibrium steady state (NESS),
provided its uniqueness. Specifically, we show how finite-time evolution in
presence of finite-sized baths, whose opportune size is determined by their
original spectral density, can be recursively used to faithfully reconstruct
the exact dynamics without requiring any small parameter. Such a reconstruction
is possible even in parameter regimes which would otherwise be inaccessible by
current state-of-the-art techniques. We specifically demonstrate this by
obtaining the full numerically exact non-Markovian dynamics of interacting
fermionic chains in two terminal set-ups with finite temperature and voltage
biases, a problem which previously remained outstanding despite its relevance
in a wide range of contexts, for example, quantum heat engines and
refrigerators.
