In this paper we consider a d=2 random Ising system on a square lattice with nearest neighbor interactions. The disorder is short range correlated and asymmetry between the vertical and the horizontal direction is admitted. More precisely, the vertical bonds are supposed to be nonrandom while the horizontal bonds alternate: one row of all nonrandom horizontal bonds is followed by one row where they are independent dichotomic random variables. We solve the model using an approximate approach that replaces the quenched average with an annealed average under the constraint that the number of frustrated plaquettes is kept fixed and equals that of the true system. The surprising fact is that for some choices of the parameters of the model there are three second-order phase transitions separating four different phases: antiferromagnetic, reentrant paramagnetic (glassy?), ferromagnetic, and paramagnetic.
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