This study develops a consensus-based transactive energy system design managed by an independent distribution system operator (DSO) for an unbalanced radial distribution network. The network is populated by welfare-maximizing customers with price-sensitive and fixed (non-price-sensitive) demands who make multiple successive power decisions during each real-time operating period OP. The DSO and customers engage in an iterative negotiation process in advance of each OP to determine retail price-to-go sequences for OP that align customer power decisions with network reliability constraints in a manner that respects customer privacy. The convergence properties of a dual decomposition algorithm developed to implement this negotiation process are analytically established. A case study is presented for an unbalanced 123-bus radial distribution network populated by household customers that demonstrates the practical effectiveness of the design.