We solve a random two-matrix model with two real asymmetric matrices whoseprimary purpose is to describe certain aspects of quantum chromodynamics withtwo colours and dynamical fermions at nonzero quark chemical potential mu. Inthis symmetry class the determinant of the Dirac operator is real but notnecessarily positive. Despite this sign problem the unquenched matrix modelremains completely solvable and provides detailed predictions for the Diracoperator spectrum in two different physical scenarios/limits: (i) theepsilon-regime of chiral perturbation theory at small mu, where mu^2 multipliedby the volume remains fixed in the infinite-volume limit and (ii) thehigh-density regime where a BCS gap is formed and mu is unscaled. We giveexplicit examples for the complex, real, and imaginary eigenvalue densitiesincluding Nf=2 non-degenerate flavours. Whilst the limit of two degeneratemasses has no sign problem and can be tested with standard lattice techniques,we analyse the severity of the sign problem for non-degenerate masses as afunction of the mass split and of mu. On the mathematical side our new results include an analytical formula forthe spectral density of real Wishart eigenvalues in the limit (i) of weaknon-Hermiticity, thus completing the previous solution of the correspondingquenched model of two real asymmetric Wishart matrices.
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