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Symbiotic radio (SR) is emerging as a spectrum- and energy-efficient communication paradigm for future passive Internet-of-things (IoT), where some single-antenna backscatter devices, referred to as Tags, are parasitic in an active primary transmission. The primary transceiver is designed to assist both direct-link (DL) and backscatter-link (BL) communication. In multi-tags SR systems, the transceiver designs become much more complicated due to the presence of DL and inter-Tag interference, which further poses new challenges to the availability and reliability of DL and BL transmission. To overcome these challenges, we formulate the stochastic optimization of transceiver design as the general network utility maximization problem (GUMP). The resultant problem is a stochastic multiple-ratio fractional non-convex problem, and consequently challenging to solve. By leveraging some fractional programming techniques, we tailor a surrogate function with the specific structure and subsequently develop a batch stochastic parallel decomposition (BSPD) algorithm, which is shown to converge to stationary solutions of the GNUMP. Simulation results verify the effectiveness of the proposed algorithm by numerical examples in terms of the achieved system throughput.