Precise coexistence curves are reported for the liquid–liquid phase transition of binary solutions of the room temperature ionic liquid (RTIL) 1-methyl-3-hexylimidazolium tetrafluoroborate (C₆mim⁺BF₄⁻) in a series of alcohols (1-butanol, 1-pentanol, 2-butanol, and 2-pentanol). The phase diagrams are determined by measuring the temperature dependence of the refractive index in the two phases of samples of critical composition. The critical data of the systems are in the region predicted for the model fluid of equal-sized, charged, hard spheres in a dielectric continuum, the so-called restricted primitive model (RPM). Therefore, the phase transition can be classified as essentially driven by Coulomb interactions. The effective exponents β_eff determined are close to the universal Ising value, where the deviations are found to be negative, when the volume fraction or the mass fraction are chosen as concentration variable. The negative values of the first Wegner correction indicate non-uniform crossover from Ising to mean-field criticality. The diameter of the coexistence curves shows the non-analytic temperature dependence typical for Ising systems.