What happened?

When grouping and reducing an array or dataset over a multi-index the coordinates that make up the multi-index get renamed to "{name_of_multiindex}_level_{i}".

It only works correctly when the Multiindex is a "homogenous grid", i.e. as obtained by stacking.

What did you expect to happen?

I expect that all coordinates keep their initial names.

Minimal Complete Verifiable Example

```Python import xarray as xr

this works:

d = xr.DataArray(range(4), dims="t", coords={"x": ("t", [0, 0, 1, 1]), "y": ("t", [0, 1, 0, 1])}) dd = d.set_index({"t": ["x", "y"]})

returns

<xarray.DataArray (t: 4)>

array([0, 1, 2, 3])

Coordinates:

* t (t) MultiIndex

- x (t) int64 0 0 1 1

- y (t) int64 0 1 0 1

dd.groupby("t").mean(...)

returns

<xarray.DataArray (t: 4)>

array([0., 1., 2., 3.])

Coordinates:

* t (t) MultiIndex

- x (t) int64 0 0 1 1

- y (t) int64 0 1 0 1

this does not work

d2 = xr.DataArray(range(6), dims="t", coords={"x": ("t", [0, 0, 1, 1, 0, 1]), "y": ("t", [0, 1, 0, 1, 0, 0])}) dd2 = d2.set_index({"t": ["x", "y"]})

returns

<xarray.DataArray (t: 6)>

array([0, 1, 2, 3, 4, 5])

Coordinates:

* t (t) MultiIndex

- x (t) int64 0 0 1 1 0 1

- y (t) int64 0 1 0 1 0 0

dd2.groupby("t").mean(...)

returns

<xarray.DataArray (t: 4)>

array([2. , 1. , 3.5, 3. ])

Coordinates:

* t (t) MultiIndex

- t_level_0 (t) int64 0 0 1 1

- t_level_1 (t) int64 0 1 0 1

```

Relevant log output

No response

Anything else we need to know?

No response

Environment

INSTALLED VERSIONS

commit: None python: 3.9.1 (default, Jan 13 2021, 15:21:08) [GCC 4.8.5 20150623 (Red Hat 4.8.5-44)] python-bits: 64 OS: Linux OS-release: 3.10.0-1160.49.1.el7.x86_64 machine: x86_64 processor: x86_64 byteorder: little LC_ALL: None LANG: en_US.UTF-8 LOCALE: ('en_US', 'UTF-8') libhdf5: 1.12.0 libnetcdf: 4.7.4

xarray: 0.21.1 pandas: 1.4.0 numpy: 1.21.5 scipy: 1.7.3 netCDF4: 1.5.8 pydap: None h5netcdf: None h5py: None Nio: None zarr: None cftime: 1.5.1.1 nc_time_axis: None PseudoNetCDF: None rasterio: None cfgrib: None iris: None bottleneck: None dask: None distributed: None matplotlib: 3.5.1 cartopy: None seaborn: None numbagg: None fsspec: None cupy: None pint: None sparse: None setuptools: 49.2.1 pip: 22.0.3 conda: None pytest: 6.2.5 IPython: 8.0.0 sphinx: None