@jononomo Your requirement that the ETS tables be perfectly synchronized is the problem - this is a non-trivial problem to solve generally, and is one that every distributed database has to deal with in some way (e.g. CAP theorem and related tradeoffs on guarantees).
Mnesia solves this problem, to some degree, though you have to do extra work to make it resilient to network partitions. Another approach is to have a process which synchronizes data between nodes, using CRDTs to handle conflict resolution, e.g. Phoenix.Tracker. There are some libraries which build on this capability, such as dispatch. You may find that the latter works better for your use case.
ETS is extremely useful for a broad array of cases, but for your specific need here, it is not the appropriate tool.
