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We construct the coordinate Bethe ansatz for all eigenstates of the openspin-$\frac12$ XXZ chain that fulfill the phantom roots criterion (PRC). Underthe PRC, the Hilbert space splits into two invariant subspaces and there aretwo sets of homogeneous Bethe ansatz equations (BAE) to characterize thesubspaces in each case. We propose two sets of vectors with chiral shocks tospan the invariant subspaces and expand the corresponding eigenstates. All thevectors are factorized and have symmetrical and simple structures. Usingseveral simple cases as examples, we present the core elements of ourgeneralized coordinate Bethe ansatz method. The eigenstates are expanded in ourgenerating set and show clear chirality and certain symmetry properties. Thebulk scattering matrices, the reflection matrices on the two boundaries and theBAE are obtained, which demonstrates the agreement with other approaches. Somehypotheses are formulated for the generalization of our approach.

Chiral coordinate Bethe ansatz for phantom eigenstates in the open XXZ spin-$\frac12$ chain