one-to-one maps on configuration space which leave the Boltzmann weight invariant. For instance, in a spin-1 antiferromagnetic chain which supports a 1D SPT known as the Haldane phase [@AKLT1988], the associated classical configuration space is one discrete variable ($m_z=-1,0,1$) per site. When viewed as an SPT phase protected by $SO(3)$ or its $Z_2 \times Z_2$ subgroup of $\pi$-rotations [@Pollmann2012], the action of the symmetry involves superpositions and cannot be considered classical. A related issue is that the microscopic mechanism stabilizing topological phases, based on matrix product states and projective symmetries [@Schuch2011], becomes obfuscated in the classical setting. Lastly, the Boltzmann weights resulting from the prescription outlined above, are complicated and anisotropic, making these models less experimentally relevant.

Interestingly, for some models based on coupled superfluids, the lattice Euclidean time partition function, following a series of transformations, can be written in a sign free manner [@Geraedts2013]. The advantage here is that the resulting models are isotropic. However in the process of making the action local, additional degrees of freedom are introduced and, from a classical perspective, it is thus unclear what are the essential ingredients which render this a well defined classical phase of matter rather than a particular model. Furthermore 