You haven't answered my question. How do nodes verify that a person making a transaction actually owns the external unit? And how do you prevent that person from spending that external unit elsewhere outside of the network? Since it's external, and the transaction only happens on the network, the unit itself hasn't actually changed hands. The original owner still possesses it, and the new owner has no way of physically obtaining it. Explain how this problem can be solved without requiring a central depository.
Well, as you even say so: stuff that is EXTERNAL to the network cannot be controlled BY the network of course.
The central depository then could be the network itself for example, as long as money is viewed as something (ultimately) physical that cannot be created by anyone within some limits though, this is going to be very difficult. If the thing transferred is pieces of data, computation or something like that, you might have better chances.