  We study spin systems defined by the winding of a random walk loop soup. For
a particular choice of loop soup intensity, we show that the corresponding spin
system is reflection-positive and is dual, in the Kramers-Wannier sense, to the
spin system $\text{sgn}(\varphi)$ where $\varphi$ is a discrete Gaussian free
field.
  In general, we show that the spin correlation functions have conformally
covariant scaling limits corresponding to the one-parameter family of functions
studied by Camia, Gandolfi and Kleban (Nuclear Physics B, 902, 2016) and
defined in terms of the winding of the Brownian loop soup. These functions have
properties consistent with the behavior of correlation functions of conformal
primaries in a conformal field theory. Here, we prove that they do correspond
to correlation functions of continuum fields (random generalized functions) for
values of the intensity of the Brownian loop soup that are not too large.
