  Manifestly non-Hermitian quantum graphs with real spectra are introduced and
shown tractable as a new class of phenomenological models with several
appealing descriptive properties. For illustrative purposes, just equilateral
star-graphs are considered here in detail, with non-Hermiticities introduced by
interactions attached to the vertices. The facilitated feasibility of the
analysis of their spectra is achieved via their systematic approximative
Runge-Kutta-inspired reduction to star-shaped discrete lattices. The resulting
bound-state spectra are found real in a discretization-independent interval of
couplings. This conclusion is reinterpreted as the existence of a hidden
Hermiticity of our models, i.e., as the standard and manifest Hermiticity of
the underlying Hamiltonian in one of less usual, {\em ad hoc} representations
${\cal H}_j$ of the Hilbert space of states in which the inner product is local
(at $j=0$) or increasingly nonlocal (at $j=1,2, ...$). Explicit examples of
these (of course, Hamiltonian-dependent) hermitizing inner products are offered
in closed form. In this way each initial quantum graph is assigned a menu of
optional, non-equivalent standard probabilistic interpretations exhibiting a
controlled, tunable nonlocality.
