A vexing problem occurs in certain flows when long, but finite, time-average estimates of the mean flow fail to exhibit the symmetry properties imposed by boundary conditions and physics. The mean field becomes suspect, making it difficult, or even incorrect to apply Reynolds decomposition. The problem occurs when the flow exhibits "super-coherent" states, i.e. states of flow having coherence times much longer than the averaging times used in typical turbulence experiments. Turbulent Rayleigh-Benard convection (RBC) is one such flow, and it will be used here as an example to illustrate and explain this phenomenon. The study focuses on a turbulent RBC experiment (Fernandes, 2001) in a 6.3:1 (diameter: depth) aspect-ratio vertical cylinder that supplemented time averaging with true ensemble averaging to achieve almost zero mean flow. To obtain a three-dimensional time-varying picture of the mechanisms at work, the experiment is simulated by direct numerical simulation of the Boussinesq equations (Sakievich et al., 2016). Three types of super-coherent states, associated with the symmetries of the flow, are found to bias the mean flow, unless steps are taken to sample each state with equal probability. They are azimuthal composition and orientation of the large-scale structures, the direction of azimuthal drift, and the preferential direction of the large-scale central motions.