We investigate the phase behavior of random copolymer melts via large-scale Monte Carlo simulations. The AB multiblock copolymers have, on average, symmetric composition and are characterized by a correlation lambda along the polymer. We employ parallel tempering and the wormhole algorithm and a technique to reduce the variance between different realizations of the disorder to explore the phase behavior. For a very large correlation of blocks, we observe a sequence of disordered phase, macrophase separation and remixing into a spatially structured (lamellar or microemulsion-like) phase upon increasing the incompatibility between different monomer species as predicted by mean field theory. For smaller values of lambda, we find that a locally segregated structure gradually forms as the incompatibility increases. As we increase the number of blocks in the polymers, the region of macrophase separation shrinks. The results of our Monte Carlo simulation are in agreement with a Ginzburg criterion, which suggests that mean field theory becomes worse as the number of blocks in a polymer increases. Different scenarios for the remixing at large incompatibility chi have been investigated. The simulation data exhibit large finite size effects. Depending on the parameters, the remixing might be either an unbinding transition, where the characteristic length scale of the spatially structured phase diverges, or a three-phase coexistence over an extended range of incompatibilities. In the latter case, the sequence distribution in the coexisting phases differs (fractionation).