Wyo. Stat. § 1-17-705
If the judgment debtor shows the court that an appeal from the foreign judgment is pending or will be taken, or that a stay of execution has been granted, and upon proof that the judgment debtor has furnished the security for satisfaction of the judgment required by the law of the state where it was rendered, the court shall stay enforcement of the foreign judgment until the appeal is concluded, the time for appeal expires, or until the stay of execution expires or is vacated. If the judgment debtor shows the court any ground upon which enforcement of a judgment of the court of this state would be stayed, the court shall stay enforcement of the foreign judgment for an appropriate period, upon requiring the security for satisfaction of the judgment which is required in this state.
W.S. 1-17-705