  In general relativity, the causal structure between events is dynamical, but
it is definite and observer-independent; events are point-like and the
membership of an event A in the future or past light-cone of an event B is an
observer-independent statement. When events are defined with respect to quantum
systems however, nothing guarantees that the causal relationship between A and
B is definite. We propose to associate a causal reference frame corresponding
to each event, which can be interpreted as an observer-dependent time according
to which an observer describes the evolution of quantum systems. In the causal
reference frame of one event, this particular event is always localised, but
other events can be "smeared out" in the future and in the past. We do not
impose a predefined causal order between the events, but only require that
descriptions from different reference frames obey a global consistency
condition. We show that our new formalism is equivalent to the pure process
matrix formalism. The latter is known to predict certain multipartite
correlations, which are incompatible with the assumption of a causal ordering
of the events -- these correlations violate causal inequalities. We show how
the causal reference frame description can be used to gain insight into the
question of realisability of such strongly non-causal processes in laboratory
experiments. As another application, we use causal reference frames to revisit
a thought experiment where the gravitational time dilation due to a massive
object in a quantum superposition of positions leads to a superposition of the
causal ordering of two events.
