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][1]. Since then, there have been several other examples, such as [this separable example][2].

The missing piece of structure is what is defined and called an *orientation* in [this article][3]. 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.


  [1]: https://www.jstor.org/stable/1970940
  [2]: https://arxiv.org/abs/math/0208084
  [3]: https://projecteuclid.org/euclid.acta/1485890059