  We investigate special solutions to the Bethe Ansatz equations (BAE) for open
integrable $XXZ$ Heisenberg spin chains containing phantom (infinite) Bethe
roots. The phantom Bethe roots do not contribute to the energy of the Bethe
state, so the energy is determined exclusively by the remaining regular
excitations. We rederive the phantom Bethe roots criterion and focus on BAE
solutions for mixtures of phantom roots and regular (finite) Bethe roots. We
prove that in the presence of phantom Bethe roots, all eigenstates are split
between two invariant subspaces, spanned by chiral shock states. Bethe
eigenstates are described by two complementary sets of Bethe Ansatz equations
for regular roots, one for each invariant subspace. The respective
"semi-phantom" Bethe vectors are states of chiral nature, with chirality
properties getting less pronounced when more regular Bethe roots are added. For
the easy plane case "semi-phantom" Bethe states carry nonzero magnetic current,
and are characterized by quasi-periodic modulation of the magnetization
profile, the most prominent example being the spin helix states (SHS). We
illustrate our results investigating "semi-phantom" Bethe states generated by
one regular Bethe root (the other Bethe roots being phantom), with simple
structure of the invariant subspace, in all details. We obtain the explicit
expressions for Bethe vectors, and calculate the simplest correlation
functions, including the spin-current for all the states in the single particle
multiplet.
