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The coupled effects of the variable-density and compressible isothermal background stratification strength on the growth of the fully compressible single-mode two-dimensional two-fluids Rayleigh--Taylor instability (RTI) are examined using direct numerical simulations (DNS) with varying Atwood numbers, A = 0.1, 0.3, and 0.5; and different background isothermal Mach numbers, Ma = 0.3, 0.9, and 1.5, respectively, in the problem Reynolds number, Re_0, range of 6375 to 51000. The results show that higher stratification strength leads to more suppression of the RTI growth for the cases with a low Atwood number. However, when the Atwood number is high, the suppression effect of compressible background stratification on the RTI growth becomes nonlinear with Ma, and in general, it becomes weaker. Furthermore, for the case with the highest background stratification strength and highest Atwood number, we observe local supersonic regions and even shock waves with increasing Re_0 at late time during the mixing. Additionally, a relevant transport equation for mixing is studied, and it is found that diffusion and production terms are dominant, and the redistribution term becomes more important with a larger Atwood number. Vortex dynamics are also analyzed using normalized vorticity and its transport equation. It is observed that for cases at various Atwood numbers, increasing Mach number generally suppresses the growth of the vortical structures. Examining the vorticity transport equation, it is shown that the baroclinicity and viscous diffusion terms are the major contributors to the change of vorticity in cases with different combinations of A and Ma. In addition, with increasing Ma, the vorticity-dilatation term becomes more significant due to the flow compressibility effects. It is also noticeable that small-scale vortical structures become more pronounced with increasing Re_0 for all Atwood numbers.