  Tracing out the environmental degrees of freedom is a necessary procedure
when simulating open quantum systems. While being an essential step in deriving
a tractable master equation it represents a loss of information. In situations
where there is strong interplay between the system and environmental degrees of
freedom this loss makes understanding the dynamics challenging. These dynamics,
when viewed in isolation, have no time-local description: they are
non-Markovian and memory effects induce complex features that are difficult to
interpret. To address this problem, we here show how to use system
correlations, calculated by any method, to infer any correlation function of a
Gaussian environment, so long as the coupling between system and environment is
linear. This not only allows reconstruction of the full dynamics of both system
and environment, but also opens avenues into studying the effect of a system on
its environment. In order to obtain accurate bath dynamics, we exploit a
numerically exact approach to simulating the system dynamics, which is based on
the construction and contraction of a tensor network that represents the
process tensor of this open quantum system. Using this we are able to find any
system correlation function exactly. To demonstrate the applicability of our
method we show how heat moves between different modes of a bosonic bath when
coupled to a two-level system that is subject to an off-resonant drive.
