If it were in a true client/server environment, where you had a variety of different client versions rolled out, then I expect you would have already found the obvious answer: the server has to support multiple versions of clients and clients use a mechanism like feature switches or capability detection to enable functionality as it becomes available server side. Both repos have their own integration tests: the server's make sure the server supports multiple client versions, and the client's make sure that the client degrades gracefully in the face of server variation.
From the way you talk, though, it sounds to me like you expect client and server to always be released at the same time (e.g., where it's a web front end and web back end). If that's really the case, then I'd just have everybody work as one team, working off a common set of features switches.
Is that helpful?