  Quantum coherence quantifies the amount of superposition in a quantum system,
and is the reason and resource behind several phenomena and technologies. It
depends on the natural basis in which the quantum state of the system is
expressed, which in turn hinges on the physical set-up being analyzed and
utilized. While quantum coherence has hitherto been conceptualized by employing
different categories of complete bases, there do exist interesting physical
situations, where the natural basis is an incomplete one, an example being an
interferometric set-up with the observer controlling only a certain fraction of
all the slits. We introduce a quantification of quantum coherence with respect
to an arbitrary incomplete basis for general quantum states, and develop the
corresponding resource theory, identifying the free states and operations.
Moreover, we obtain a complementarity relation between the so-defined quantum
coherence and the which-path information in an interferometric set-up with
several slits, of which only a section is in control of the observer or is
accessible to her. This therefore provides us with another face of the
wave-particle duality in quantum systems, demonstrating that the
complementarity is functional in more general set-ups than thus far considered.
