I will present a stochastic analysis of the sine-Gordon Euclidean quantum field (cos(β φ))_2 on the full space up to the second threshold, i.e. for β^2<6*π. The basis of our method is a stochastic quantisation equation given by a forward-backward stochastic differential equation (FBSDE) for a decomposition (X_t)_(t≥0) of the interacting Euclidean field X_∞ along a scale parameter t≥0 using an approximate version of the renormalisation flow equation. The FBSDE produces a scale-by-scale coupling of the interacting field with the Gaussian free field without cut-offs and describes the optimiser of a stochastic control problem for Euclidean QFTs introduced by Barashkov and Gubinelli. I will first explain the general set-up for the FBSDE approach. In the case of the sine-Gordon model, I will showcase some applications of the FBSDE to illustrate that it can be used effectively to obtain results about large deviations, integrability, decay of correlations for local observables, singularity with respect to the free field, Osterwalder-Schrader axioms and other properties. This is joint work with Massimiliano Gubinelli.