  Understanding the causal influences that hold among parts of a system is
critical both to explaining that system's natural behaviour and to controlling
it through targeted interventions. In a quantum world, understanding causal
relations is equally important, but the set of possibilities is far richer. The
two basic ways in which a pair of time-ordered quantum systems may be causally
related are by a cause-effect mechanism or by a common cause acting on both.
Here, we show a coherent mixture of these two possibilities. We realize this
nonclassical causal relation in a quantum optics experiment and derive a set of
criteria for witnessing the coherence based on a quantum version of Berkson's
effect, whereby two independent causes can become correlated upon observation
of their common effect. The interplay of causality and quantum theory lies at
the heart of challenging foundational puzzles, including Bell's theorem and the
search for quantum gravity.
