Multiple DC microgrids (DCMGs) are interconnected to form the DCMG cluster that can achieve various benefits of microgrid-based power solutions, e.g., low cost, high-demand response, high resilience, etc. The task to manage power flow that enables the benefits of DCMG clusters is assigned to the tertiary control in the hierarchical control paradigm. Meanwhile, the tertiary control is usually implemented with distributed communication which can enhance fault-tolerance, reliability, and expandability of DCMG clusters. So far, the existing distributed tertiary control is in general based on the conventional linear proportional-integral (PI) algorithms, and thus the performance of the linear controls for such complex nonlinear and higher-order networks of the DCMG clusters are limited. This paper introduces prediction function control (PFC) for the tertiary level of the hierarchical control of the DCMG clusters and implements the new algorithm with remained distributed control structure. In the proposed distributed predictive tertiary control (DPTC), the dynamic consensus protocol is deployed based on the sharing of information between neighboring agents. Under various test scenarios, the effectiveness of the DPTC strategy for the DCMG cluster that consists of three islanding DCMGs is verified through simulations and experimental results.