Specifications vs tests

This is partly a rant, but also to get a gauge of how other organizations approach this aspect of design & development.

The boss loves TDD. Nothing wrong with that, though I've found it can slow down development at the beginning of new projects. I'm rewriting an old service that was written poorly in haste. Fairly common in the software world. I've asked for a spec document numerous times, but the boss won't write it. S/he'd rather brain dump and write tests. Tests are great as a backstop, but they're not specs. My preferred workflow is to start with the specs, then write the tests and the code separately against those specs. Instead, the boss says that the tests are the spec, and when there are any vagaries, compare it against the old service. They've admitted that the old service has bugs, so it's frustrating and surprising that they'd point to that as the reference.

TBH I'm starting to lose faith in this organization. What would you do or say to change it?