With the help of detailed atomistic simulations, we have determined the phase boundary of a complex coacervate system resulting from the complexation of two oppositely and fully charged weak polyelectrolytes, namely, poly(acrylic acid) (PAA) and poly(N,N-dimethylaminoethyl methacrylate) (PDMAEMA), characterized by degrees of polymerization N = 20 and 17, respectively, in an aqueous KCl solution. The binodal phase behavior was computed through an iterative procedure of balancing the total chemical potentials of water and salt in the supernatant and coacervate phases. Prior to any phase equilibria computations, several free energy calculation methods were evaluated in terms of their capability to provide accurate predictions of the excess chemical potential of water and salt at different temperatures. Methods based on free energy perturbation and thermodynamic integration were found to be the most reliable. Representative points on the salt-polymer binodal phase diagram were found to be in remarkably good agreement with the experimental data, as well as with a previous coarse-grained molecular dynamics (MD) study. We also studied the salt partitioning by directly simulating the two phases in contact one with the other at equilibrium, which showed a clear preference of salt ions for the polymer-lean phase; the predicted tie-lines exhibit negative slopes, denoting that the salt concentration in the supernatant phase is higher than in the coacervate phase. Additional results from the atomistic simulations regarding the pairs of atoms in the two oppositely charged polyelectrolytes that make the most important contributions to the complexation are also presented and discussed. Overall, our study suggests that fully detailed atomistic MD simulations can be successfully employed to predict with remarkable accuracy the phase behavior and salt partitioning of coacervate-based systems, thus offering significant insights into the complexation phenomenon and its underlying physics.