Microstructure evolution in complex nonlinear systems is of great interest from both scientific and engineering viewpoints. Here, we consider an important case of such an evolution, a coherent decomposition of a homogeneous parent phase involving the symmetry-lifting crystal lattice rearrangement of the product phase. It is shown that under certain conditions the transformation develops as a pseudospinodal decomposition, which is defined as a transformation with continuous changes of the compositions of both decomposed phases towards their equilibrium values. The pseudospinodal decomposition starts by the formation of a nanodomain precursor state with a so-called tweed structure. The three-dimensional modelling demonstrates that this tweed structure has an underlying chessboard feature, and this arrangement is a template for the further microstructure development that eventually produces a coherent two-phase chessboard nanowire pattern. The proposed theory and modelling describe the mechanism of the pseudospinodal decomposition and predict all observed three-dimensional features of the chessboard structure.