  Causality can be defined in terms of space-time or based on
information-theoretic structures, which correspond to very different notions of
causation. Yet, in physical experiments, these notions play together in a
compatible manner. The process matrix framework is useful for modelling
indefinite causal structures (ICS) in an information-theoretic sense, but there
remain important open questions regarding the physicality of such processes. In
particular, there are several experiments that claim to implement ICS processes
in Minkowski space-time, which presents an apparent theoretical paradox: how
can an indefinite information-theoretic causal structure be compatible with a
definite space-time structure? To address this, we develop a general framework
that disentangles the two causality notions and formalises their relations. The
framework describes a composition of quantum operations through feedback loops,
and the embedding of the resulting (possibly cyclic) information-theoretic
structure in an acyclic space-time structure. Relativistic causality is
formalised as an operational compatibility condition between the two
structures. Reformulating the process matrix framework here, we establish no-go
results which imply that it is impossible to physically realise ICS in a fixed
space-time with space-time localised quantum systems. Further, we prove that
physical realisations of any ICS process, even those involving space-time
non-localised systems, will ultimately admit an explanation in terms of a
definite (and acyclic) causal order process, at a more fine-grained level.
These results fully resolve the apparent paradox and we discuss their
implications for the interpretation of the above-mentioned experiments.
Moreover, our work offers concrete insights on the operational meaning of
indefinite causality, both within and beyond the context of a fixed space-time.
