  Many-body localised phases of disordered, interacting quantum systems allow
for exotic localisation protected quantum order in eigenstates at arbitrarily
high energy densities. In this work, we analyse the manifestation of such order
on the Hilbert-space anatomy of eigenstates. Quantified in terms of non-local
Hilbert-spatial correlations of eigenstate amplitudes, we find that the spread
of the eigenstates on the Hilbert-space graph is directly related to the order
parameters which characterise the localisation protected order, and hence these
correlations, in turn, characterise the order or lack thereof. Higher-point
eigenstate correlations also characterise the different entanglement structures
in the many-body localised phases, with and without order, as well as in the
ergodic phase. The results pave the way for characterising the transitions
between many-body localised phases and the ergodic phase in terms of scaling of
emergent correlation lengthscales on the Hilbert-space graph.
