What is the dual-write problem, and why can't a naive database-update-then-network-call approach avoid it?

If a database update succeeds but a subsequent, separate network call to notify another service fails (a crash, a network partition), the local data is committed but the other service never finds out — the two systems silently diverge. This is unavoidable with this approach because the database commit and the network call are two entirely separate operations with no shared transactional guarantee.