This paper proposes a real-time distributed operational architecture to coordinate integrated transmission and distribution systems (ITD). At the distribution system level, the distribution system operator (DSO) calculates the aggregated flexibility of all controllable devices by power-energy envelopes and provides them to the transmission system operator (TSO). At the transmission system level, a distributed nonlinear model predictive control (nmpc) approach is proposed to coordinate the economic dispatch of multiple TSOs, considering the aggregated flexibility of all distribution systems. The subproblems of the proposed approach are associated with different TSOs and individual time periods. In addition, the aggregated flexibility of controllable devices in distribution networks is encapsulated, re-calculated, and communicated through the power-energy envelopes, facilitating a reduction in computational complexity and eliminating redundant information exchanges between TSOs and DSOs, thereby enhancing privacy and security. The framework's effectiveness and applicability in real-world scenarios are validated through simulated operational scenarios on a summer day in Germany, highlighting its robustness in the face of significant prediction mismatches due to severe weather conditions.