Multi-Carrier Energy Systems (MCESs) have attracted global appeal and indicated broad outlooks in developing green energy resources in the community. The transactive networked operation of multiple MCESs improves economic opportunities and system flexibility. Nevertheless, how to fairly and convincingly allocate the total expenses among MCESs is a crucial challenge that is addressed in this paper. To tackle the fairness issue, a two-stage optimization framework in the transactive networked MCESs is developed. In the first stage, named the islanding operation mode, the operation cost of each MCES is minimized and the optimization outcomes are sent to the central operator of the transactive networked MCESs. In the second stage, the central operator solves an optimization problem to minimize the total operation cost of the transactive networked MCESs considering energy transactions among MCESs and fairness constraints. The fair cost allocation constraints lead to similar or less contributions in total operation cost in the transactive networked structure compared with the islanding mode. The proposed model is structured as a Mixed-Integer Linear Programming problem and formulated in GAMS modeling language. The results indicate that the proposed two-stage optimization approach decreases the operation cost of the networked MCESs by 16.7% while the fairness issue was satisfied.