We study the basic properties of tensor random fields (TRFs) of the wide-sense homogeneous and isotropic kind with generally anisotropic realizations. Working within the constraints of small strains, attention is given to antiplane elasticity, thermal conductivity, classical elasticity and micropolar elasticity, all in quasi-static settings albeit without making any specific statements about the Fourier and Hooke laws. The field equations (such as linear and angular momentum balances and strain–displacement relations) lead to consequences for the respective dependent fields involved. In effect, these consequences are restrictions on the admissible forms of the correlation functions describing the TRFs.