The Martin Kleppmann suggestion is great - his book is excellent as well.

I work on a similar problem where we have distributed devices running Elixir and Nerves that have intermittent connectivity, but need to reliably transmit data to and from the cloud. The approach I ended up taking was very similar to Kafka or database replication, where data is written to a sequential log on either end and the log is asynchronously replicated between client and server with at least once delivery. This approach solves many problems (reliable delivery, easy to audit or reconstruct system state, etc.), but introduces new ones (linearizability, immutable data, write amplification, etc.). Building on top of this you can use CRDT type approaches like a PN Counter to make eventually consistent values. Beware, though, distributed systems can be a real pain in the neck.

The system runs in production quite well and there is an example app here, however I am sharing this mostly for educational purposes since the protocol is still changing I’m not sure how much support I can give.

It’s a really interesting problem, though, and I’m happy to let you bounce any ideas you have off of me.

Subscriptions, listen and notify all require network and wouldn’t help you when the user is offline. Offline-syncing is very much depends on complexity of your data structures, how conflicts are to be resolved, whether you want to only push offline writes from a single client per object (easy), or you need to be able to resolve conflicts between multiple clients to the same object (hard), etc. I’m not aware of any off-the-shelf solutions in elixir ecosystem. There are some CRDT and the recently released Delta which seems to be OT libs out there but any of them might be not suitable in depending on your usecase.

Off-topic – there are really good lectures on youtube by Martin Kleppmann on this and related topics.