Fiat currency backs the military, and the military backs the fiat currency.
You are describing dollar not fiat in general. Dollar is forced on the rest of the world using military force.
Otherwise other fiat currencies are backed by actual economy of the countries. For example the fiat currencies the ASEAN members are using amongst themselves is not being forced on each member using military. Or the Yuan that China is getting more countries to use (PetroYuan is a thing now!) is being backed by the strong Chinese economy not its military.