The detailed structure of singularities of ordered field represents a fundamental problem in diverse areas of physics. At the defect cores, the deformations are so strong that the system explores states with symmetry different from that of an undistorted material. These regions are difficult to explore experimentally as their spatial extension is very small, a few molecular lengths in the condensed matter. Here we explore the cores of disclinations in the so-called chromonic nematics that extend over macroscopic length scales accessible for optical characterization. We demonstrate that the amplitude S and the phase (the director) of the order parameter vary along both the radial and azimuthal directions, in contrast to the classic models in which S varies only with the distance from the centre and depends only on the azimuthal coordinate. This unexpected core structure is explained by a strong coupling of the phase and amplitude of the order parameter in the free energy.