Soft tissues carry out their vital biological functions within a dynamic mechanical framework that can be extended or compressed. Externally or internally applied uni-axial or biaxial changes induce longitudinal strains that can be of either sign. The complex interrelationship between applied strain and induced strain is quantified by a time-space change of the Poisson’s ratio, which is itself determined by cell–cell and cell–matrix interactions. While the viscoelasticity of multicellular systems under various experimental conditions has already been discussed extensively, the role of the viscoelastic Poisson’s ratio, as a vital indicator of tissue compressibility, is only now beginning to be appreciated and explored more thoroughly. Tissues have frequently been treated as incompressible. However, the porous structure of the cell membranes, tissues, and extracellular matrices ensures an outflow of liquid even under relatively modest physiological strain conditions. This study explores a range of tissues and biological composites consisting of multiple cell types and extracellular matrices in the context of compressibility, accompanied by their Poisson’s ratio. They are subjected to strains induced by both external and internal factors that mimic physiological conditions.