Unique factorization of based anti-automorphisms

For any ring $A$ and index type $I$, suppose $A$ has a $\mathbb Z$-basis $(b_i)_{i\in I}$, a distinguished index $e$ with $b_e=1$, and a $\mathbb Z$-linear involution $x\mapsto\bar x$. Assume all coefficients $[b_i b_j]_k$ are nonnegative, $\overline{xy}=\bar y\,\bar x$ for all $x,y\in A$, every $\bar b_i$ is a basis element, and $[b_i b_j]_m=[b_m\bar b_j]_i$ for all $i,j,m\in I$, where brackets denote basis coefficients. Given any additive bijection $f:A\to A$ with $f(1)=1$, $f(xy)=f(y)f(x)$ for every $x,y\in A$, and $\{f(b_i):i\in I\}=\{b_i:i\in I\}$, there exists exactly one based ring automorphism structure $g$: its underlying map is a unital ring isomorphism $g:A\to A$ satisfying $\{g(b_i):i\in I\}=\{b_i:i\in I\}$, and it obeys $f(a)=\overline{g(a)}$ for every $a\in A$. Uniqueness is among all such based ring automorphism structures satisfying this equation.