  A recent framework of quantum theory with no global causal order predicts the
existence of "causally nonseparable" processes. Some of these processes produce
correlations incompatible with any causal order (they violate so-called "causal
inequalities" analogous to Bell inequalities) while others do not (they admit a
"causal model" analogous to a local model). Here we show for the first time
that bipartite causally nonseparable processes with a causal model exist, and
give evidence that they have no clear physical interpretation. We also provide
an algorithm to generate processes of this kind and show that they have nonzero
measure in the set of all processes. We demonstrate the existence of processes
which stop violating causal inequalities but are still causally nonseparable
when mixed with a certain amount of "white noise". This is reminiscent of the
behavior of Werner states in the context of entanglement and nonlocality.
Finally, we provide numerical evidence for the existence of causally
nonseparable processes which have a causal model even when extended with an
entangled state shared among the parties.
