A key problem in the description of non-ideal, multi-component plasmas is the drastic difference in the characteristic length and time scales of the different particle species. This challenging multiscale problem inherent to studying streaming complex plasmas can efficiently be tackled by a statistical ansatz for the light plasma constituents in combination with first-principle Langevin dynamics simulations of the heavy and strongly correlated dust component. Of crucial importance in this scheme is the quality of the dynamically screened Coulomb potential. For this purpose, we introduce Kielstream, a new high-performance computer code for the computation of three-dimensional plasma wakefields and the resulting electric fields. The optimization techniques used and the handling of competing numerical errors are discussed in detail. Results are presented for the wakefield around a single dust grain as well as multiscale simulations of a correlated ensemble of grains revealing fundamental structural changes when wake effects take charge.