We study finite real hyperfields with the cyclic positive cones. We present an algorithm fordetermining all such hyperfields up to isomorphism and compute its C-characteristic. By a computerprogramming we give a complete classification of finite real hyperfields of orders up to 13 and determinewhich C-characteristics appear in such hyperfields with orders up to 17. In this database we present the numerical results concerning our algorithm in the case when thepositive cone is a cyclic group of order N , for N ≤ 8.