Is there a system that ensures global bank transfers net to zero?

There isn’t an institution per se, there are many, depending on how the payments are being cleared. BIS (Bank of International Settlements) based in Basel, Switzerland is one. There is the CLS Group out of NY that pretty much handles the worlds FX, and then there are many direct/indirect correspondent banking relationships which ensure that double-entry accounting in the ledger of the various banks involved is there. Each bank also maintains an account ledger with the corresponding bank also. Thus if you screw up on your accounting, the corresponding set would not match.

There used to be a lot of abuse of this system when ledgers were offline and it took days to synchronize them, but not anymore. Now, ledger entries are cross-posted and ledger balances are easily balanced. If each country is defined as a pyramid, with the central bank being on top (as an example) and banks and other financial institutions going down the pyramid, then at consolidation time, everything funnels up the pyramid and it is very easy to then balance the entries of the day. The tops of the pyramid then are also balanced with one other (correspondent banking so to speak).