For an asymmetric sinh-Poisson problem arising as a mean field equation of equilibrium turbulence vortices with variable intensities of interest in hydrodynamic turbulence, we address the existence of bubbling solutions on compact Riemann surfaces. By using a Lyapunov–Schmidt reduction, we find sufficient conditions under which there exist bubbling solutions blowing up at m different points of S: positively at m1 points and negatively at m − m1 points with m ≥ 1 and m1 ∈ {0, 1,…,m}. Several examples in different situations illustrate our results in the sphere S2 and flat two-torus T including non-negative potentials with zero set non-empty.