  Quantum coherence plays an important role in quantum information protocols
that provide an advantage over classical information processing. The amount of
coherence that can exist between two orthogonal subspaces is limited by the
positivity constraint on the density matrix. On the level of multipartite
systems, this gives rise to what is known as monogamy of entanglement. On the
level of single systems this leads to a bound, and hence, a trade-off in
coherence that can exist between different orthogonal subspaces. In this work
we derive trade-off relations for the amount of coherence that can be shared
between a given subspace and all other subspaces based on trace norm,
Hilbert-Schmidt norm and von Neumann relative entropy. From this we derive
criteria detecting genuine multisubspace coherence.
