Described simply, double-entry bookkeeping allows firms to maintain records that reflect what the firm owns and owes and also what the firm has earned and spent over any given period of time. Double-entry bookkeeping revolutionized the field of financial accounting during the Renaissance period. Whereas simple ledgers had long been the standard for record keeping for merchants, the church and state treasuries, the growth of long distance trade and creation of the first joint stock companies resulted in firms whose records were too voluminous and complicated to provide any assurance of accuracy to their users.
Triple entry accounting is an enhancement to the traditional double-entry system in which all accounting entries involving outside parties are cryptographically sealed by a third entry. These include purchases of inventory and supplies, sales, tax and utility payments and other expenses. Placed side by side, the bookkeeping entries of both parties to a given transaction are congruent. A seller books a debit to account for cash received, while a buyer books a credit for cash spent in the same transaction, but in separate sets of accounting records. This is where the blockchain comes in: rather than these entries occurring separately in independent sets of books, they occur in the form of a transfer between wallet addresses in the same distributed, public ledger, creating an interlocking system of enduring accounting records. Since the entries are distributed and cryptographically sealed, falsifying them in a credible way or destroying them to conceal activity is practically impossible.
Source and further reading: Triple-Entry Bookkeeping, Bitcoin Magazine
In practice, for a UK solution, it is envisaged that two or more of the UK's Big Four accounting & audit practice firms would be the validators on a permissioned distributed ledger used to process and record triple entry accounting records and provide trusted real-time auditing and "Proof of Transaction."
The term "Triple Entry Accounting," was first used by Ian Grigg, financial cryptographer, and described in his paper published in 2005, three years before the emergence of Bitcoin and its underlying Blockchain protocol.