In this scheme the ecash notes don't represent IOUs, the spent ecash notes represent the liability of the User towards the Mint
Bitcoin denominated ecash credit secured by publicly arbitrated escrow providing symmetric trust/incentive relationship between Mint and User. Works similar to a credit card top-up scheme. Instead of depositing bitcoin to the Mint to get ecash issued, the Mint issues credit in the form of ecash. The spent ecash tokens represent the User's liability towards the Mint. The User must periodically provide proof of the unspent balance (turning in expired tokens, which can no longer be spent) and settle the difference by paying a Lightning Network invoice to get a new batch of tokens issued.
Lightning Network interoperable Chaumian ecash implemented on bitcoin provides excellent privacy, but many have complained about the Mint's ability to rugpull the Users. There is also an increased regulatory risk concerning custodians, that provide anonymous payment services.
The Bank/Mint extends you credit in the form of ecash, and the Users spend that.
The User and the Mint form a multi-sig escrow with a specific amount, but that TXO is not touched. The User gets ecash notes/tokens issued each month. When they expire, the Users have to turn in their remaining notes and settle the outstanding balance over lightning or by other means, to get a new batch of ecash tokens issued. It's like any p2p escrow solution.
Dispute resolution can be arbitrarily complicated and decentralized. Reputable trusted arbitration Oracle Services may arise (that don't actually custody user funds so they are not a financial provider) could slip under the radar. Typically the User and the Mint would agree on a set of trusted Arbiters to handle dispute over the escrow.
When opening an account the User will deposit the limit + penalty amount into the escrow contract. After that limit amount of ecash is issued to the User. Limits would typically be selected from a standard set of amounts to increase the anonymity set. Before funding the 2-of-2 escrow contract, both parties sign and retain a transaction that spends it over to the arbitration contract. SIGHASH SINGLE|ANYONECANPAY ensures that the escrow input and arbitration output are signed but fees can be paid by additional inputs. This allows for the initiating party to pay the current feerate at the time.
Spending and Receiving works exactly the same as with custodial ecash.
To re-up the balance first the User will ask for a re-up invoice for the spent amount in the previous month. This will give him a full limit amount of ecash tokens/notes. Then the User turns these tokens in by signing an authentication message with his escrow key. This does not allow the Mint to learn how much of the balance each User actually spent, as they always turn in a full balance, unless they wish to close the account and cooperatively settle on-chain.
When the User is not able to present the exact full balance amount, the User will have to pay interest on the outstanding sum. For example this interest could be 5%. Same thing happens if the User has too much ecash from the previous month, the User pays a fine which is again 5% of the surplus amount. After deducting the fine or interest from the presented amount, the User is issued new ecash notes in the remaining sum.
Alternatively instead of getting new ecash tokens issued, the User may request an on-chain settlement with the Mint and thus closing the account. In case of a dispute or non-cooperation a presigned transaction will spend the 2-of-2 escrow to the arbitration contract (for example 3-of-4 with the 2 Oracles), which can have relative timelocks on exit conditions. In case an Oracle Service goes out of business the Mint and the User can always sign a transaction to a new arbitration contract, with a new set of Oracles they can agree on.
This is an interesting way to introduce credit into the system while minimizing credit risk.