The ground state of a rotating Bose-Einstein condensate trapped in a two-dimensional anharmonic--anisotropic potential is analyzed numerically at the limit of an infinite number of particles. We find that the density breaks up along the $x$ direction in position space and along the $p_y$ direction in momentum space together with the acquisition of angular momentum. Side by side, the anisotropies of the many-particle position variances along the $x$ and $y$ directions and of the many-particle momentum variances along the $p_y$ and $p_x$ directions become opposite when computed at the many-body and mean-field levels of theory. All in all, the rotating bosons are found to possess unique correlations at the limit of an infinite number of particles, both in position and momentum spaces, although their many-body and mean-field energies per particle and densities per particle coincide and the condensate fraction is 100\%. Implications are briefly discussed.