This paper considers a RIS-assisted symbiotic communication system, where additional information is conveyed by the passive reconfigurable intelligent surface (RIS). In existing schemes, individual phase modulation is usually adopted at the RIS elements, which severely limits exploiting all extra multiplexing gains brought by the RIS. To address the issue, we propose a novel matrix decomposition algorithm that transforms the equivalent channel into a structured form while effectively suppressing the decomposition residual. Based on this, a novel transceiver architecture employing grouped annulus modulation (GAM) with a hexagonal-lattice-based constellation is developed, which is capable of achieving the full degrees of freedom (DoFs) when the decomposition algorithm performs as expected. Numerical results demonstrate that the proposed transceiver achieves much higher communication rates, thereby leading to higher spectral efficiency, compared to the conventional phase-only modulation scheme, while maintaining comparable error performance.