of the two phases change. Considering bulk physics, long range order in $\rho$ implies the following non local (string) order parameter in the non-trivial phase: $$\begin{aligned}
  \label{eq:sAs}
   \left< \rho_i \rho_j \right> &=\left< \sigma_i\prod_{\ell \in \Gamma_{ij}}U_{\ell} \sigma_j\right> 
  \to \text{const}\, \end{aligned}$$ as $\text{dist}(i,j)\to\infty$ and $\Gamma_{ij}$ is a path from $i$ to $j$. Alternatively stated, performing the non-local transformation $\sigma_i \rightarrow \rho_i=\prod_{\ell\in\Gamma_{0i}}U_\ell
\sigma_i$, with $0$ a reference site, unveils a hidden ferromagnetic phase for the non-trivial order, whereas for the trivial phase, this results in a simple paramagnet.

As we now argue the hidden ferromagnetic order is a distinguishing property of the topological phase and therefore one cannot continuously deform the models onto one another. This implies that there are at least two distinct admissible phases in our classification for $d=2,G=G'=Z_2$. Notably local and symmetric perturbations in the original $U$ and $\sigma$ variables would be transformed into local and symmetric perturbations in $\mu$ and $\sigma$. As this transformation has no effect on the free energy, one finds that hidden order is thermodynamically equivalent to conventional order. This means that hidden order not just a feature of the model but rather a robust property which can only vanish through a phase transition or by 