A cloud-native application should consist of one or more stateless processes, with any persisted data being stored using a backing service. The processes of an application following the twelve-factor app methodology are stateless and do not share anything. Adhering to this rule will allow your application to be more robust and to scale more easily.
For example, in order for a web application to be a cloud-native application, it should not rely on sticky sessions. Sticky sessions are where a router/load balancer can assign a single server to a particular user so that subsequent requests from the same user will be routed to the same server. Among other things, this could allow caching user session data in the memory of the application's process, with the expectation that subsequent requests from the same user will be routed to the same process and have access to the cached data. Instead of using sticky sessions, a cloud-native application should cache by some other means, such as using a distributed caching system like Redis or Memcached.