  We theoretically predict, and experimentally verify with entangled photons,
that outcome communication is not enough for hidden-state models to reproduce
quantum steering. Hidden-state models with outcome communication correspond, in
turn, to the well-known instrumental processes of causal inference but in the
1-sided device-independent (1S DI) scenario of one black-box measurement device
and one well-characterised quantum apparatus. We introduce 1S-DI instrumental
inequalities to test against these models, with the appealing feature of
detecting entanglement even when communication of the black box's measurement
outcome is allowed. We find that, remarkably, these inequalities can also be
violated solely with steering, i.e. without outcome communication. In fact, an
efficiently-computable formal quantifier -- the robustness of
non-instrumentality -- naturally arises; and we prove that steering alone is
enough to maximize it. Our findings imply that quantum theory admits a stronger
form of steering than known until now, with fundamental as well as practical
potential implications.
