  The quantum many-body bound-state problem in its computationally successful
coupled cluster method (CCM) representation is reconsidered. In conventional
practice one factorizes the ground-state wave functions $|\Psi\rangle= e^S
|\Phi\rangle$ which live in the "physical" Hilbert space ${\cal H}^{(P)}$ using
an elementary ansatz for $|\Phi\rangle$ plus a formal expansion of $S$ in an
operator basis of multi-configurational creation operators. In our paper a
reinterpretation of the method is proposed. Using parallels between the CCM and
the so called quasi-Hermitian, alias three-Hilbert-space (THS), quantum
mechanics, the CCM transition from the known microscopic Hamiltonian (denoted
by usual symbol $H$), which is self-adjoint in ${\cal H}^{(P)}$, to its
effective lower-case isospectral avatar $\hat{h}=e^{-S} H e^S$, is assigned a
THS interpretation. In the opposite direction, a THS-prescribed, non-CCM,
innovative reinstallation of Hermiticity is shown to be possible for the CCM
effective Hamiltonian $\hat{h}$, which only appears manifestly non-Hermitian in
its own ("friendly") Hilbert space ${\cal H}^{(F)}$. This goal is achieved via
an ad hoc amendment of the inner product in ${\cal H}^{(F)}$, thereby yielding
the third ("standard") Hilbert space ${\cal H}^{(S)}$. Due to the resulting
exact unitary equivalence between the first and third spaces, ${\cal
H}^{(P)}\sim {\cal H}^{(S)}$, the indistinguishability of predictions
calculated in these alternative physical frameworks is guaranteed.
