My main reference for learning about Eisenstein series has been section 3.7 of Bump's "Automorphic forms and representations", so I'll try to use the notation from there. These ideas are relatively new to me, and I think that I might be misunderstanding something important.
Let $\chi_1$ and $\chi_2$ be characters of the ideles $A^\times$ of some number field $F$. Then the principal series representation $\pi(\chi_1,\chi_2)$ of $GL(2,A)$ is a restricted tensor product of the local representations $\pi(\chi_{1,v},\chi_{2,v})$, and the local representations are irreducible whenever the $\chi_i$ are both unramified at $v$. The Eisenstein series then defines a $GL(2,A)$-equivariant map $\pi(\chi_1,\chi_2)\rightarrow \mathcal{A}(GL(2,F)\backslash GL(2,A),\omega)$ into the space of automorphic forms with central character $\omega=\chi_1 \chi_2$; an element $f\in\pi(\chi_1,\chi_2)$ is sent to the Eisenstein series $E(g,f)$ considered as function of $g\in GL(2,A)$.
Suppose further that $\chi_1$ and $\chi_2$ are not unitary but $\pi(\chi_1,\chi_2)$ is. Then $\chi_1=\xi_1|\cdot|^s$ for some real non-zero $s$ and unitary $\xi_1$, so the $\chi_{1,v}$ are not unitary. The image of the above map into $\mathcal{A}(GL(2,F)\backslash GL(2,A),\omega)$ is isomorphic to some quotient of $\pi(\chi_1,\chi_2)$ and contains an irreducible subrepresentation $V$. The local representations $V_v$ in its tensor product decomposition are then subquotients of the $\pi(\chi_{1,v},\chi_{2,v})$ (I think; is this correct?). So for unramified $v$ these are zero or $\pi(\chi_{1,v},\chi_{2,v})$. But assuming the Ramanujan conjecture, the $V_v$ are tempered when the $\chi_{i,v}$ are unramified, so the latter possibility does not occur.
So $V$ is in fact zero and all of the Eisenstein series $E(g,f)$ for $f\in \pi(\chi_1,\chi_2)$ are zero. This would imply that the only non-zero Eisenstein series for $f$ in a unitary principal series representation are those for $f$ in a principal series representation with both $\chi_i$ unitary.
Later on in section 3.7, letting $f$ vary in way that the $\chi_i$ depend on $s\in\mathbb{C}$ and most of the $f$ are in principal series representations with $\chi_i$ definitely not unitary, an analytic continuation for $E(g,f)$ as a meromorphic function of $s$ is established. In particular the $E(g,f)$ are not all zero, and the $\chi_i$ can be chosen to depend on $s$ while keeping each $\pi(\chi_1,\chi_2)$ unitary. Where did the first argument go wrong?