We model an experimentally plausible implementation of a synthetic RNA-based biochemical toggle switch proposed in previous work by the authors. We show that the system structure is suited to exhibit multistationarity for arbitrary choice of the parameters. The network is based on in vitro transcription and nucleic acid strand displacement reactions. It is composed of two distinct RNA polymerases producing mutually inhibiting RNA aptamers. The aptamer inhibitors create an overall positive loop, where regulation is achieved by modulating the activity of the polymerases rather than the promoter activity. Inhibition occurs via stoichiometric binding of RNA monomers to enzymes and is not a cooperative phenomenon; the only nonlinearities in the differential equations are given by second order reaction rates. Enzyme activity is recovered in the presence of DNA strands that displace the aptamers from their target, and mediate their degradation; recovery is also a bimolecular binding process. Numerical analysis shows that the system admits bistability in a wide range of parameters.