Based on the consolidation solutions of cross‐anisotropic stratified fractional visco‐poroelastic soils, the interaction between an irregular rigid foundation and the underlying soils in the time domain is studied. Firstly, the foundation is discretized into a series of elements, which is similar to the division principle of the finite element method (FEM). The displacement equations of the foundation elements are established according to the fact that the rigid foundation remains flat when loaded. Secondly, the flexibility matrix of the soils is obtained from the consolidation solution, which is taken as the kernel function of the Boundary element method (BEM). After that, the interaction equation is derived according to the displacement compatibility condition and equilibrium of forces between the rigid foundation and the soils. So, the relation between the displacement and the contact force of the soils is attained. Finally, the rotation, displacement, and reaction of the foundation are obtained by solving the interaction equation. The results are compared with several published references to prove the validity of this study. Furthermore, numerical examples are designed to investigate the effects of the viscoelasticity, cross‐anisotropy, stratification, aspect ratio and the irregular shape on the time‐varying interaction.