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In the present paper, we study the Bethe ansatz equations for Gaudin modeland Miura opers in characteristic $p>0$. Our study is based on a work by E.Frenkel, in which solutions to the Bethe ansatz equations are described interms of Miura opers on the complex projective line. The main result of thepresent paper provides a positive characteristic analogue of this description.We pay particular attention to the case of Miura $\mathrm{PGL}_2$-opers becausedormant generic Miura $\mathrm{PGL}_2$-opers correspond bijectively to Tangostructures, which bring various sorts of pathological phenomena in positivecharacteristic (e.g., counter-examples to the Kodaira vanishing theorem). As aconsequence, we can construct new examples of Tango structures by means ofsolutions to the Bethe ansatz equations modulo $p$.

Gaudin model modulo $p$, Tango structures, and dormant Miura opers