The effect of random anisotropy on the magnetic structure (the pattern of directions of the antiferromagnetic vector) is studied in planar antiferromagnets. Large uniform domains can appear only due to random anisotropy fields with a correlation radius exceeding a domain wall dimension. The fields with a smaller correlation radius can generate only an amorphous structure with the antiferromagnetic vector varying continuously everywhere in the crystal volume. It is shown how parameters of magnetic structures may be extracted from experimental data of magnetic torque and broadening of AFMR linewidths. The data for AuMn, CsMnF₃ and Cr are discussed. Interpretation of magnetic torque experiments in the transverse spin density wave phase of the antiferromagnetic Cr increases the scale (the polarisation domain size) of the magnetic structure 30 times compared to the scale obtained from thermal activation theory. Because of random anisotropy of random fields, an intermediate 'phase' appears in the process of the phase transition near the critical temperature of the perfect crystal.