including the interaction force will result in a non-ideal equation of state describing the fluid. Dependent on this equation of state, you can then determine critical properties (critical density, pressure, ‘temperature’). And in the single component multiphase version of the shan chen model, the parameter G corresponds to a temperature-like variable.
To have a liquid and a vapor phase develop, you have to pick a value for G that leads to a temperature below the critical one (see Yuan & Schaefer: Phys. Fluids 18, 042101 (2006)). The farther you are below the critical temperature, the larger the density ratio of vapor and liquid phase will be.
Now how to pick G? Usually people want a desired density ratio between vapor and liquid. So a function for Psi is picked (which is stable at the desired density ratio) which will result in a specific equation of state. To get an estimate of the equilibrium densities at different temperatures either the Maxwell construction or the Mechanical stability condition (Shan: Phys. Rev. E 77, 066702 (2008)) can be solved for the particular equation of state. In the end, G (the tempereature) is chosen to get the desired equilibrium densities.