We experimentally investigate the highly modulated turbulent wake behind a wall-mounted square-base pyramid protruding through the boundary layer. We present the first modal energy flow analysis of a time-resolved three-dimensional velocity field from experimental particle image velocimetry data. The underlying low-order representation is optimized for resolving the base-flow variation as well as the first and second harmonics associated with vortex shedding – generalizing the triple decomposition of Reynolds & Hussain (J. Fluid Mech., vol. 54, 1972, pp. 263–288). This analysis comprises not only a detailed modal balance of turbulent kinetic energy as pioneered by Rempfer & Fasel (J. Fluid Mech., vol. 275, 1994, pp. 257–283) for proper orthogonal decomposition (POD) models, but also the companion energy balance of the mean flow. The experimental results vividly demonstrate how constitutive elements of mean-field theory (Stuart, J. Fluid Mech., vol. 4, 1958, pp. 1–21) near laminar Hopf bifurcations remain strongly pronounced in a turbulent wake characterized by highly modulated, quasi-periodic shedding. The study emphasizes, for instance, the stabilizing role of mean-field manifolds, as explored in the pioneering POD model of Aubry et al. (J. Fluid Mech., vol. 192, 1988, pp. 115–173). The presented low-order representation of the flow and modal energy flow analyses may provide important insights and reference data for computational turbulence modelling, e.g. unsteady Reynolds-averaged Navier–Stokes simulations.