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In the present paper, a numerical investigation was performed to assess the effect of the rheological behavior of non-Newtonian fluids on Rayleigh–Bénard thermosolutal convection instabilities within shallow and finite aspect ratio enclosures. Neumann and Dirichlet thermal and solutal boundary condition types were applied on the horizontal walls of the enclosure. Using the Boussinesq approximation, the momentum, energy, and species transport equations were numerically solved using a finite difference method. Performing a nonlinear asymptotic analysis, a bistability convective phenomenon was discovered, which was induced by the combined fluid shear-thinning and aiding thermosolutal convection effects. Therefore, bistability convection was the main focus in the current study using the more practical constitutive Carreau–Yasuda viscosity model, which is valid from zero to infinite shear rates. Also, the combined effects of the rheology parameters and double diffusive bistability convection were studied. For aiding flow, the shear-thinning and the slower diffusing solute effects were counteracting and, as a result, two steady-state finite amplitude solutions were found to exist for the same values of the governing parameters, which indicated and demonstrated evidence for the existence of bistability convective flows. For opposing flows, the shear-thinning effect strengthened subcritical flows, which sustained well below the threshold of Newtonian thermosolutal convection. Thus, bistability convection did not exist for opposing flows, as both the shear-thinning and the slower diffusing component effects favored subcritical convection.