  We develop an extension of the process matrix (PM) framework for correlations
between quantum operations with no causal order that allows multiple rounds of
information exchange for each party compatibly with the assumption of
well-defined causal order of events locally. We characterise the higher-order
process describing such correlations, which we name the multi-round process
matrix (MPM), and formulate a notion of causal nonseparability for it that
extends the one for standard PMs. We show that in the multi-round case there
are novel manifestations of causal nonseparability that are not captured by a
naive application of the standard PM formalism: we exhibit an instance of an
operator that is both a valid PM and a valid MPM, but is causally separable in
the first case and can violate causal inequalities in the second case due to
the possibility of using a side channel.
