There is a policy that mypy never infers Any in joins of precise types. So this is more related to #3830, in the sense should we instead infer List[Sized] in this case? On one hand I would say we can, but on the other hand this is quite tricky in general (see #3257) and therefore low priority. You can just use explicit annotations in such cases:

o: List[Sized] = [[1, 2], ["foo", "bar"]]

Thanks for your investigation, I don't have any legitimacy to discuss your policies, from an external eye it seems too bad to prefer the inference of List[object] over the correct (and imprecise) List[List[Any]].
You're right it would also be sufficient in this case to have a List[Sized], but it would still be incorrect and could lead to false positives with things like:

o: List[Sized] = [[1, 2], ["foo", "bar"]]
[x.count(1) for x in o]

I will try your suggestion to use explicit type annotation :)

The workaround gets a little tedious when your problematic list is being defined as part of a dictionary. You still have to explicitly assert its type after the fact:

assert isinstance(foo["o"], list)

This is a won't fix. mypy's inference should never produce Any. It would be expensive for mypy to construct ad hoc protocols and any time you create a type that a user cannot easily name it creates usability issues