We revisit the theory of tunnel-coupled atomic quasicondensates in double-well elongated traps at finite temperatures. Using the functional-integral approach, we calculate the relative-phase correlation function beyond the harmonic limit of small fluctuations of the relative phase and its conjugate relative-density variable. We show that the thermal fluctuations of the relative phase between the two quasicondensates decrease the frequency of Josephson oscillations and even wash out these oscillations for small values of the tunnel coupling.