For the massless sine-Gordon model at the free fermion point, in infinite volume, we define the fractional (charge or vertex operator) correlation functions from the probabilistic path integral and prove that they are given by renormalized determinants of massive twisted Dirac operators. The fractional correlation functions are the moments of the imaginary multiplicative chaos, a random generalized function that we construct with respect to the infinite-volume massless sine-Gordon measure. The renormalized determinants are the tau functions of Sato--Miwa--Jimbo as identified by Palmer. The construction and a priori control of the imaginary multiplicative chaos combines methods from stochastic analysis (of singular SPDE flavor) for short-scale regularity with qualitative input from integrability for large-scale control. The exact identification of the correlation functions with the renormalized determinants relies on finite-volume approximation, regularity estimates for the mass perturbation, and analytic continuation in the coupling constant. The combination of existing results for tau functions with our identification implies various predictions for the sine-Gordon model such as that the fractional two-point functions are expressed as Fredholm determinants and satisfy certain PDEs as predicted by Bernard--LeClair. Using asymptotics of Fredholm determinants of Basor--Tracy and mixing of the massless sine-Gordon model at the free fermion point, which we prove, we further derive the exact formula for the one-point function predicted by Lukyanov--Zamolodchikov (at the free fermion point).