We address the problem of cyclone-anticyclone asymmetry in geophysical turbulence using a direct numerical simulation with high Reynolds number Re similar to 15 000 that includes an active upper boundary and interior dynamics. The regime, characterized by a finite Rossby number (Ro similar to 0.6) strongly departs from the classical quasigeostrophic regime. The numerical resolution is pushed to the limit of today's supercomputer capabilities ensuring more than two decades free of viscous effects. The results show a strong cyclonic dominance in the upper layers that is stronger for filaments than for vortices. This is in contrast with similar studies that have no active upper boundary which reported either anticyclone dominance or a symmetry between cyclones and anticyclones in the upper layers. This highlights the impact of boundary dynamics on geophysical turbulence.