Acoustoelasticity describes the interaction of acoustic waves with nonlinear elastic deformations, particularly the change of wave velocity due to initial stresses or strains in a predeformed body. The theory extends the strain energy to cubic terms (third-order elasticity) and allows for finite strains to model deformations at high confining pressures. However, the theory considers equant (stiff) pores but neglects the effects of soft (compliant) pores, such as microfractures, cracks, and grain contacts. Our main contribution is to include these effects. Application of the novel poroacoustoelasticity theory to ultrasonic measurements on carbonate samples at varying confining pressures provides a better fit for the measured data of pressure dependence of wave velocity. We have quantified the contribution of the compliant pores to the nonlinear behavior of the wave velocity and determined the relation between the threshold pressure (beyond which the theories with and without compliant pores yield the same velocity) and porosity and permeability. The extension of poroacoustoelasticity theory by incorporating a dual-pore structure provides better description for stress dependence of wave velocity in fluid-saturated heterogeneous rocks, which can be applicable in further field studies regarding reservoir characterization and in situ stress estimation.