html_url,issue_url,id,node_id,user,created_at,updated_at,author_association,body,reactions,performed_via_github_app,issue
https://github.com/pydata/xarray/issues/6969#issuecomment-1233148389,https://api.github.com/repos/pydata/xarray/issues/6969,1233148389,IC_kwDOAMm_X85JgFnl,4160723,2022-08-31T16:16:26Z,2022-08-31T16:16:26Z,MEMBER,"Thanks for the report @bboutanquoi.

The issue seems to be related to `set_index()`, which in your example does not convert back the old multi-index coordinates back to regular variables:

```python
y = y.set_index(multi=['sub_id', 'last_j'])
type(y.composite_id.variable)
# IndexVariable (should be Variable)
```

I guess it's similar to #6946 which is related to `reset_index`. 


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