We propose a duality analysis for obtaining the critical manifold of two-dimensional spin glasses. Our method is based on the computation of quenched free energies with periodic and twisted periodic boundary conditions on a finite basis. The precision can be systematically improved by increasing the size of the basis, leading to very fast convergence towards the thermodynamic limit. We apply the method to obtain the phase diagrams of the random-bond Ising model and $q$-state Potts gauge glasses. In the Ising case, the Nishimori point is found at $p_N = 0.10929 pm 0.00002$, in agreement with and improving on the precision of existing numerical estimations. Similar precision is found throughout the high-temperature part of the phase diagram. Finite-size effects are larger in the low-temperature region, but our results are in qualitative agreement with the known features of the phase diagram. In particular we show analytically that the critical point in the ground state is located at finite $p_0$.
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