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What happened:

When I try to assign an instance of pd.MultiIndex to the coords: the behavior of "Dimension coordinate" is to maintain the multi-index so I can use multi-index levels directly as keyword arguments, while the behavior of "Non-dimension coordinate" is to change the index to an np.ndarray with dtype "object" , that make above function failed.

What you expected to happen:

I want the "Non-dimension coordinate" can also maintain the multi-index

Minimal Complete Verifiable Example:

```python

import xarray as xr import numpy as np import pandas as pd

create a dataarray named 'arr'

arr = xr.DataArray(np.r_[:6],{},'z')

add coords 'z1' in dim 'z'

this time 'z1' is a "Non-dimension coordinate"

arr.coords['z1'] = 'z',pd.MultiIndex.from_product( ([1,2,3],['a','b']),names=['i','n'] )

let the coords 'z1' to be the z's coords

arr.set_index(z='z1') arr.sel(n='a') # Failed to use index 'n' in multiindex 'z' ValueError: dimensions or multi-index levels ['n'] do not exist

let's see what's coords 'z' look like

arr.z <xarray.DataArray 'z' (z: 6)> array([(1, 'a'), (1, 'b'), (2, 'a'), (2, 'b'), (3, 'a'), (3, 'b')], dtype=object) Coordinates: * z (z) object (1, 'a') (1, 'b') (2, 'a') (2, 'b') (3, 'a') (3, 'b')

why 'z' is not a MultiIndex ???

now 'z' is a "Dimension coordinate"

set the coords again

arr1.coords['z'] = pd.MultiIndex.from_product( ([1,2,3],['a','b']),names=['i','n'] ) arr.sel(n='a') #OK <xarray.DataArray (i: 3)> array([0, 2, 4]) Coordinates: * i (i) int64 1 2 3

let's see what's coords 'z' look like

arr.z <xarray.DataArray 'z' (z: 6)> array([(1, 'a'), (1, 'b'), (2, 'a'), (2, 'b'), (3, 'a'), (3, 'b')], dtype=object) Coordinates: * z (z) MultiIndex - i (z) int64 1 1 2 2 3 3 - n (z) object 'a' 'b' 'a' 'b' 'a' 'b'

'z' is successfully setted to a MultiIndex

```

Anything else we need to know?:

Environment:

Output of <tt>xr.show_versions()</tt>

INSTALLED VERSIONS ------------------ commit: None python: 3.8.2 (default, Mar 25 2020, 17:03:02) [GCC 7.3.0] python-bits: 64 OS: Linux OS-release: 5.10.27-gentoo-dist machine: x86_64 processor: byteorder: little LC_ALL: None LANG: C.UTF-8 LOCALE: en_US.UTF-8 libhdf5: 1.10.4 libnetcdf: None xarray: 0.17.0 pandas: 1.2.4 numpy: 1.20.2 scipy: 1.6.2 netCDF4: None pydap: None h5netcdf: None h5py: 2.10.0 Nio: None zarr: None cftime: 1.4.1 nc_time_axis: None PseudoNetCDF: None rasterio: None cfgrib: 0.9.8.5 iris: None bottleneck: None dask: 2021.04.0 distributed: 2021.04.0 matplotlib: 3.4.1 cartopy: 0.18.0 seaborn: 0.11.1 numbagg: None pint: None setuptools: 49.6.0.post20210108 pip: 21.0.1 conda: 4.10.1 pytest: None IPython: 7.22.0 sphinx: None