  Quantum contextuality describes situations where the statistics observed in
different measurement contexts cannot be explained by a measurement independent
reality of the system. The most simple case is observed in a three-dimensional
Hilbert space, with five different measurement contexts related to each other
by shared measurement outcomes. The quantum formalism defines the relations
between these contexts in terms of well-defined relations between operators,
and these relations can be used to reconstruct an unknown quantum state from a
finite set of measurement results. Here, I introduce a reconstruction method
based on the relations between the five measurement contexts that can violate
the bounds of non-contextual statistics. A complete description of an arbitrary
quantum state requires only five of the eight elements of a Kirkwood-Dirac
quasi probability, but only an overcomplete set of eleven elements provides an
unbiased description of all five contexts. A set of five fundamental relations
between the eleven elements reveals a deterministic structure that links the
five contexts. As illustrated by a number of examples, these relations provide
a consistent description of contextual realities for the measurement outcomes
of all five contexts.
