You can get a very basic real-time push in PostgreSQL with per-row triggers and NOTIFY/LISTEN (pg_notify/pg_listen).
The hard part is building out a change feed that lets you synchronize on a subset of a table in a way that is performant, and to make this work with joins (i.e. when you're using the database as more than just a document store).
The hard part is building out a change feed that lets you synchronize on a subset of a table in a way that is performant, and to make this work with joins (i.e. when you're using the database as more than just a document store).