  Any quantum resource theory is based on free states and free operations,
i.e., states and operations which can be created and performed at no cost. In
the resource theory of coherence free states are diagonal in some fixed basis,
and free operations are those which cannot create coherence for some particular
experimental realization. Recently, some problems of this approach have been
discussed, and new sets of operations have been proposed to resolve these
problems. As an example for such a problem, a quantum operation can be
incoherent in one particular experimental realization, but it still can create
large amount of coherence in another. This problem is resolved by the framework
of genuine quantum coherence. Here, we investigate this concept further and
study other more general sets of operations which cannot create coherence in
any experimental realization. We analyze in detail the mathematical structure
of this class of maps and use this to study possible state transformations
under these operations. We show that deterministic manipulation is severely
limited, even in the asymptotic settings. In particular, this framework does
not have a unique golden unit, i.e., there is no single state from which all
other states can be created deterministically with the free operations. On the
other hand, we show that these theories allow for a richer structure in what
comes to stochastic manipulation of states. In particular, we determine the
optimal probability of conversion among pure states with genuinely incoherent
operations. Still, the aforementioned limitations suggest that any reasonably
powerful resource theory of coherence must contain free operations which can
potentially create coherence in some experimental realization.
