The obvious suggestion would be to have internal groups (excluding legal) collaborate on some common plaintext markdown format where DVCS tools would provide an advantage, then publish to docx and hand over to legal for review. Unfortunately what always happens is that major changes to the contract are needed after legal has done their work, and it's easier (and acceptable) for all groups to work on the final paper together. That's where it breaks down.
I really feel that in 2016 having some poor bastard do clerical coordination of this stuff is just wrong. It's a mechanical task that's highly both error prone and a major source of operational risk, and it should be doable by the computer.
I actually kind of want to build this. Word (and the rest of office, I have the same problems with spreadsheets and powerpoints) plus plugins as the IDE, solid version control in the backend.