The federated are given a kind of btc paper called fm-sats.

When someone from the federation wants to pay a merchant outside of it, what he does is "sell" the corresponding amount of fm-sats, and the federation sends the merchant real sats.

It is impossible to control that the federations do not issue more fm-sats than the collateral they have, then secondary markets for fm-sats are organized and thus we already have much more paper sats than real sats.

Exactly what the banks did in the days of the gold standard.