  We introduce the idea of weakly coherent collisional models, where the
elements of an environment interacting with a system of interest are prepared
in states that are approximately thermal, but have an amount of coherence
proportional to a short system-environment interaction time in a scenario akin
to well-known collisional models. We show that, in the continuous-time limit,
the model allows for a clear formulation of the first and second laws of
thermodynamics, which are modified to include a non-trivial contribution
related to quantum coherence. Remarkably, we derive a bound showing that the
degree of such coherence in the state of the elements of the environment
represents a resource, which can be consumed to convert heat into an ordered
(unitary-like) energy term in the system, even though no work is performed in
the global dynamics. Our results therefore represent an instance where
thermodynamics can be extended beyond thermal systems, opening the way for
combining classical and quantum resources.
