  We construct the coordinate Bethe ansatz for all eigenstates of the open
spin-$\frac12$ XXZ chain that fulfill the phantom roots criterion (PRC). Under
the PRC, the Hilbert space splits into two invariant subspaces and there are
two sets of homogeneous Bethe ansatz equations (BAE) to characterize the
subspaces in each case. We propose two sets of vectors with chiral shocks to
span the invariant subspaces and expand the corresponding eigenstates. All the
vectors are factorized and have symmetrical and simple structures. Using
several simple cases as examples, we present the core elements of our
generalized coordinate Bethe ansatz method. The eigenstates are expanded in our
generating set and show clear chirality and certain symmetry properties. The
bulk scattering matrices, the reflection matrices on the two boundaries and the
BAE are obtained, which demonstrates the agreement with other approaches. Some
hypotheses are formulated for the generalization of our approach.
