I've been toying with the idea of writing a protocol for Git as a "Blockchain" for bank interchange. Require signatures on all commits, include a protocol for how to push commit proposals to other peers for signing, verifying commits before they're merged, etc. No mining, just a distributed transaction log via git.
Merged where? If you have the analog of github in your scheme, then you have a central authority. If you have a central authority, you don't need a blockchain [for banking], not really.
If you assume fully independent git nodes that interact, then there is no authoritative 'branch', 'repo', or anything. Every node will have it's own world view (repo and history) and to enforce a 'global consensus' you will need a BFT distributed consensus algorithm on top of your git scheme so all bank can agree on transactions and apply patches (accept transactions).
If you assume partial consensus -- only interacting banks of a transaction need to be in agreement -- then you still need a consensus for that smaller group to 'merge' their collective actions into a linear transitional narrative.
For most people using 'git', the "distributed transaction log via git" is owned and managed by GitHub, a central authority.
It's a generic distrubuted data structure in git, with identities and signature, and conflict merge. At the moment it's used to store bugs, more to come later.
