REGR: MultiIndex.join not resorting levels of new index
- [x] closes #53093 (Replace xxxx with the GitHub issue number)
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [x] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [x] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

Not resorting causes all kinds of problems when indexing
Cleaned up, we should aim at fixing this properly for 2.1, because you can still end up like this if you instantiate the MultiIndex directly Owee, I'm MrMeeseeks, Look at me.

There seem to be a conflict, please backport manually. Here are approximate instructions:

1. Checkout backport branch and update it.

```
git checkout 2.0.x
git pull
```

2. Cherry pick the first parent branch of the this PR on top of the older branch:
```
git cherry-pick -x -m1 ddff84a5ddbf55b5f407976be359720804cd6981
```

3. You will likely have some merge/cherry-pick conflict here, fix them and commit:

```
git commit -am 'Backport PR #53102: REGR: MultiIndex.join not resorting levels of new index'
```

4. Push to a named branch:

```
git push YOURFORK 2.0.x:auto-backport-of-pr-53102-on-2.0.x
```

5. Create a PR against branch 2.0.x, I would have named this PR:

> "Backport PR #53102 on branch 2.0.x (REGR: MultiIndex.join not resorting levels of new index)"

And apply the correct labels and milestones.

Congratulations — you did some good work! Hopefully your backport PR will be tested by the continuous integration and merged soon!

Remember to remove the `Still Needs Manual Backport` label once the PR gets merged.

If these instructions are inaccurate, feel free to [suggest an improvement](https://github.com/MeeseeksBox/MeeseeksDev).
                 @meeseeksdev backport 2.0.x