Cloud virtualization technology is shifting towards light-weight containers, which provide isolated environments for running cloud-based services. The emerging trends such as container-based micro-service architectures and hybrid cloud deployments result in increased traffic volumes between the micro-services, mobility of the communication endpoints, and some of the communication taking place over untrusted networks. Yet, the services are typically designed with the assumption of scalable, persistent and secure connectivity. In this paper, we present the SynAPTIC architecture, which enables secure and persistent connectivity between mobile containers, especially in the hybrid cloud and in multi-tenant cloud networks. The solution is based on the standardized Host Identity Protocol (HIP) that tenants can deploy on top of existing cloud infrastructure independently of their cloud provider. Optional cloud-provider extensions based on Software-Defined Networking (SDN) further optimize the networking architecture. Our qualitative and quantitative evaluation shows that SynAPTIC performs better than some of the existing solutions.