Modular Multilevel Converters (MMCs) in HVDC applications usually adopt a distributed control architecture to manage a large number of sub-modules (SMs) through a communication network. The communication congestion and network disconnection might lead to communication interruption (CI) and eventually cause the system to malfunction. In this paper, a resilient operation strategy is proposed and studied to ride through the CI fault, in order to prevent frequent fault SM bypassing, replacement, or even system shutdown. The analysis of the MMC distributed control system with the presence of CI indicates that the insertion index of the faulted SM might become constant, which distorts the output current and results in over-voltage of the communication interrupted SM (CI-SM). The CI-SM capacitor voltage prediction can be used to determine the MMC safe operation period after CI occurs. During the safe operation period, the CI-SM power balance is sustained by utilizing pre-stored phase signals to generate a sinusoidal insertion index according to its capacitor voltage tracking error. Two operation modes are proposed and analyzed to ensure the MMC stable operation under various conditions. The system protection is sensibly used only if the CI duration exceeds a safe operation period, which avoids frequent SM cut-off. Good agreement of the CI-SM capacitor voltage is achieved between the theoretical and simulation results. The effectiveness and robustness of the proposed MMC resilient operation are experimentally confirmed.