This paper presents an algorithm for computing minimal ordered Coxian representations of phase-type distributions whose Laplace-Stieltjes transform has only real poles. We first identify a set of necessary and sufficient conditions for an ordered Coxian representation to be minimal with respect to the number of phases involved. The conditions establish a relationship between the Coxian representations of a Coxian distribution and the derivatives of its distribution function at zero. Based on the conditions, the algorithm is developed. Three numerical examples show the effectiveness of the algorithm and some geometric properties associated with ordered Coxian representations.