Wyo. Stat. Ann. § 1-17-206

When a court renders judgment against two (2) or more persons, any of whom are sureties for another in the contract on which the judgment is founded, there shall be no stay of execution on the judgment if the sureties object at the time of rendering the judgment unless the surety for the stay of execution will undertake specially to pay the judgment in case the amount cannot be levied of the principal defendant.