We present a discontinuous Galerkin formulation of the consistent splitting scheme introduced in [1]. The method enforces incompressibility implicitly via a pressure Poisson equation, which removes velocity-pressure compatibility constraints and eliminates pressure boundary layers. By imposing consistent boundary conditions, the temporal accuracy is not limited by splitting errors. With suitable choices of numerical fluxes in the L2-conforming discretization, the consistent boundary condition reduces to contributions involving only the acceleration and viscous terms. Applying Leray projection to control the divergence error in the velocity field further simplifies the boundary condition to depend only on data from the current time step. For time discretization, a high-order BDF scheme is applied. By adjusting the extrapolation order of the convective term on the right-hand side of the pressure Poisson equation, the maximum admissible time step size for BDF-3 and BDF-4 schemes is extended compared to the original formulation in [1]. Numerical experiments demonstrate that this modification does not deteriorate the temporal convergence rate. The convective term is treated in a semi-implicit manner, which permits larger CFL numbers than explicit approaches while avoiding the nonlinear systems associated with fully implicit formulations. To increase accuracy per degree of freedom, high-order polynomial approximations are employed. Combined with matrix-free operator evaluations, these choices result in an efficient and scalable algorithm suitable for modern high-performance computing architectures. The proposed method is validated on standard benchmark problems for incompressible flows, including flow past a cylinder and the three-dimensional Taylor–Green vortex. Its performance is assessed through comparisons with a fully coupled solver approach and the higher-order dual splitting scheme in terms of stability and time to solution, demonstrating robustness and suitability for large-scale real-world applications.