For any C$^*$-algebra $A$, we can define its opposite algebra $A^{\mathrm{op}}$, which is the algebra where $ab$ is defined to be $ba$, as calculated in $A$. Let's restrict to unital algebras for simplicity. Then the identity mapping $A \rightarrow A^{\mathrm{op}}$ is linear, positive and unital, and is its own inverse. Therefore it induces a continuous, affine, origin-preserving isomorphism of quasi-state spaces. To get a counterexample, all you need is a C$^*$-algebra that is not isomorphic to its opposite. As far as I know the first example was a von Neumann algebra constructed by Connes. Since then, there have been several other examples, such as this separable example.
The missing piece of structure is what is defined and called an orientation in this article. The second author of that article is sometimes here on this forum. If you want to apply that article to the non-unital case, just remember that the quasi-state space of a C$^*$-algebra $A$ is isomorphic to the state space of the unitization $\tilde{A}$ in the obvious manner.