The numerical simulation of compressible turbulent flows in complex geometries represents a challenge for several aerospace applications, especially for configurations with very high values of Reynolds number. The use of body fitted meshes requires particular care in the mesh generation step when strongly anisotropic cells are generated near wall in the presence of complex geometric features. Among the different immersed boundary strategies that have been proposed to avoid body fitted meshes, the Shifted Boundary Method (SBM) [1] allows to enforce boundary conditions on a surrogate (shifted) surface rather than on body-fitted boundaries. In this way, the method alleviates several long-standing practical and numerical obstacles: the high cost and fragility of mesh generation for complex geometries, element distortion and loss of accuracy near walls, boundary-layer overlap problems that arise with high-order curvilinear elements, and the demanding cut-cell quadrature typically required by classical immersed-boundary techniques. In this work, the SBM was implemented in the framework of a discontinuous Galerkin (DG) discretisation for compressible flow simulations. The proposed SBM–DG formulation enforces boundary conditions weakly on surrogate surfaces: this is achieved by introducing local corrections to the gradients of the conservative variables within the DG surface-integral terms, thereby consistently incorporating the effect of the true geometry while operating on meshes that do not conform to solid walls. Numerical evidence indicates that the SBM–DG strategy delivers encouraging accuracy and stability for compressible regimes, while retaining the flexibility and locality that make DG attractive for complex physics and adaptive workflows. Several numerical tests have been carried out to investigate different correction strategies in order to minimize oscillations in pressure and skin friction at wall. Importantly, the formulation is shown to be effective within a Reynolds-Averaged Navier–Stokes (RANS) framework, suggesting its suitability for engineering-grade turbulent-flow computations.