In Alzheimer's disease, neurofibrillary lesions correlate with cognitive deficits and consist of inclusions of tau protein with cross-beta structure. A stable dimeric form of soluble tau has been evidenced in the cells, but its high-resolution structure is missing in solution. We know, however, that cryo-electron microscopy (c-EM) of full-length tau in the brain of an individual with AD displays a core of eight beta-sheets with a C-shaped architecture spanning the R3-R4 repeat domain, while the rest of the protein is very flexible. To address the conformational ensemble of the dimer, we performed atomistic replica exchange molecular dynamics simulations on the tau R3-R4 domain starting from the c-EM configuration. We find that the wild type tau R3-R4 dimer explores elongated, U-shaped, V-shaped, and globular forms rather than the C-shape. Phosphorylation of Ser356, pSer356, is known to block the interaction between the tau protein and the amyloid-beta 42 peptide. Standard molecular dynamics simulations of this phosphorylated sequence for a total of 5 mu s compared to its wild type counterpart show a modulation of the population of beta-helices and accessible topologies and a decrease of intermediates near the fibril-like conformers.