  Recent frameworks describing quantum mechanics in the absence of a global
causal order admit the existence of causally indefinite processes, where it is
impossible to ascribe causal order for events A and B. These frameworks even
allow for processes that violate the so-called causal inequalities, which are
analogous to Bell's inequalities. However, the physicality of these exotic
processes is, in the general case, still under debate, bringing into question
their foundational relevance. While it is known that causally indefinite
processes can be probabilistically realised by means of a quantum circuit,
along with an additional conditioning event C, concrete insights into the
ontological meaning of such implementation schemes have heretofore been
limited. Here, we show that causally indefinite processes can be realised with
schemes where C serves only as a classical flag heralding which causally
indefinite process was realised. We then show that there are processes where
any pure conditioning measurement of C leads to a causally indefinite process
for A and B, thus establishing causal indefiniteness as a basis-independent
quantity. Finally, we demonstrate that quantum mechanics allows for phenomena
where C can deterministically decide whether A comes before B or vice versa,
without signalling to either. This is akin to Wheeler's famous delayed-choice
experiment establishing definite causal order in quantum mechanics as
instrument-\textit{dependent} property.
