In this paper, we propose a novel risk-limiting real-time pricing mechanism for a regional electricity network of prosumers with distributed solar power generation. The target system is a regional electricity network that consists of many prosumers, each equipped with a battery and a renewable energy-based generator. In particular, we assume that each prosumer is in possession of a solar power generator, i.e., photovoltaic cells. In the proposed market mechanism, each prosumer sets an asking price and bids for the amount of electricity that he/she sells or buys, In addition, he/she sets an asking price and bids for a parameter to cope with the uncertainty induced by solar power generation in a day-ahead market, as well. It is a characteristic of regional electricity networks of prosumers that there is a strong correlation in the solar power generation of each house within a network. A market mechanism for a regional electricity network of prosumers with distributed solar power generation should consider such characteristics. The performance of the proposed mechanism is demonstrated numerically through a simulation experiment. It outperformed a conventional real-time pricing method that does not consider uncertainty, and a method that does not involve trading of the parameter for coping with uncertainty.