Compression stiffening, or an increase in shear modulus with increasing compressive strain, has been observed in recent rheometry experiments on brain, liver, and fat tissues. Here we extend the known types of biomaterials exhibiting this phenomenon to include agarose gel and fruit flesh. The data reveal a linear relationship between shear storage modulus and uniaxial prestress, even up to 40% strain in some cases. We focus on this less-familiar linear relationship to show that two different results from classic elasticity theory can account for the phenomenon of linear compression stiffening. One result is due to Barron and Klein, extended here to the relevant geometry and prestresses; the other is due to Birch. For incompressible materials, there are no adjustable parameters in either theory. Which one applies to a given situation is a matter of reference state, suggesting that the reference state is determined by the tendency of the material to develop, or not develop, axial stress (in excess of the applied prestress) when subjected to torsion at constant axial strain. Our experiments and analysis also strengthen the notion that seemingly distinct animal and plant tissues can have mechanically similar behavior at the quantitative level under certain conditions.