It seems to me that there may be another reason why (logarithmic) confinement of defects due to Coulombic interactions should not be regarded as "topological order at nonzero temperature", irrespective of how if it realized by an underlying microscopic Hamiltonian.

Consider the toric code on a plane with two large punctures. We can encode a qubit in the sector with zero total electric charge and zero total magnetic charge: the encoded |0> is the state with nontrivial Z2 electric charge and trivial Z2 magnetic charge in each hole, and the encoded |+> is the state with nonzero Z2 magnetic charge and trivial Z2 electric charge in each hole. The qubit is protected because the energy splitting becomes exponentially small as the radius r of each hole gets large and the distance R between the holes also gets large.

If we now introduce a Coulombic attractive interaction between defects, there is a contribution to the qubit's energy that scales like log(R/r) and depends on the charge. This might be acceptable if the energy splitting were very stable, but in fact the energy depends on the details of the geometry of the hole's boundary. Interactions with the bath that excite fluctuations of the boundary can drive decoherence of the qubit; it is not topologically protected.

This may be a fundamental problem. On the one hand, we want infinite-range interactions between defects to prevent logical errors due to long-distance diffusion of the defects. On the other hand, infinite-range interactions induce nontopological contributions to the qubit's energy, interfering with topological protection.

Stronger confinement of the defects would not seem to improve the situation. In either the gapped Higgs phase (with linearly confined magnetic charges and unconfined electric defects) or the gapped confinement phase (linearly confined electric charges and unconfined magnetic defects) there is no surviving topological order and hence no topological quantum memory.

And yet ... as Barbara says we might regard the scheme with active error correction (in which the defects are periodically identified and paired to prevent the formation of long error chains that cause logical errors) as an existence proof for 2D topological memory that can be stabilized at nonzero temperature. This scheme is realizable in principle and we have nice ways to model it. Yes .. but it is much different than stabilizing the topological memory with Coulombic pairwise interactions between defects. Isn't it?