  While the standard formulation of quantum theory assumes a fixed background
causal structure, one can relax this assumption within the so-called process
matrix framework. Remarkably, some processes, termed causally nonseparable, are
incompatible with a definite causal order. We explore a form of certification
of causal nonseparability in a semi-device-independent scenario where the
involved parties receive trusted quantum inputs, but whose operations are
otherwise uncharacterised. Defining the notion of causally nonseparable
distributed measurements, we show that certain causally nonseparable processes
which cannot violate any causal inequality, including the canonical example of
the quantum switch, can generate noncausal correlations in such a scenario.
Moreover, by imposing some further natural structure to the untrusted
operations, we show that all bipartite causally nonseparable process matrices
can be certified with trusted quantum inputs.
