From 3478da73a5912f67c8d6c19496fb306e9f26d493 Mon Sep 17 00:00:00 2001
From: Benoit Bovy <benbovy@gmail.com>
Date: Fri, 26 Jan 2024 11:38:18 +0100
Subject: [PATCH 1/3] fix serialize multi-index level coord after reset

---
 xarray/conventions.py | 18 +++++++++++-------
 1 file changed, 11 insertions(+), 7 deletions(-)

diff --git a/xarray/conventions.py b/xarray/conventions.py
index 1d8e81e1bf2..61d0cc5e385 100644
--- a/xarray/conventions.py
+++ b/xarray/conventions.py
@@ -16,7 +16,7 @@
 )
 from xarray.core.pycompat import is_duck_dask_array
 from xarray.core.utils import emit_user_level_warning
-from xarray.core.variable import Variable
+from xarray.core.variable import IndexVariable, Variable
 
 CF_RELATED_DATA = (
     "bounds",
@@ -84,13 +84,17 @@ def _infer_dtype(array, name=None):
 
 
 def ensure_not_multiindex(var: Variable, name: T_Name = None) -> None:
+    # only the pandas multi-index dimension coordinate cannot be serialized (tuple values)
     if isinstance(var._data, indexing.PandasMultiIndexingAdapter):
-        raise NotImplementedError(
-            f"variable {name!r} is a MultiIndex, which cannot yet be "
-            "serialized. Instead, either use reset_index() "
-            "to convert MultiIndex levels into coordinate variables instead "
-            "or use https://cf-xarray.readthedocs.io/en/latest/coding.html."
-        )
+        if name is None and isinstance(var, IndexVariable):
+            name = var.name
+        if var.dims == (name,):
+            raise NotImplementedError(
+                f"variable {name!r} is a MultiIndex, which cannot yet be "
+                "serialized. Instead, either use reset_index() "
+                "to convert MultiIndex levels into coordinate variables instead "
+                "or use https://cf-xarray.readthedocs.io/en/latest/coding.html."
+            )
 
 
 def _copy_with_dtype(data, dtype: np.typing.DTypeLike):

From 82ca94e9d0ed36922e9d301a1c24243e57b69239 Mon Sep 17 00:00:00 2001
From: Benoit Bovy <benbovy@gmail.com>
Date: Fri, 26 Jan 2024 11:39:32 +0100
Subject: [PATCH 2/3] add regression test

---
 xarray/tests/test_backends.py | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/xarray/tests/test_backends.py b/xarray/tests/test_backends.py
index 85afbc1e147..03d2565d65e 100644
--- a/xarray/tests/test_backends.py
+++ b/xarray/tests/test_backends.py
@@ -1245,6 +1245,11 @@ def test_multiindex_not_implemented(self) -> None:
             with self.roundtrip(ds):
                 pass
 
+        # regression GH8628 (can serialize reset multi-index level coordinates)
+        ds_reset = ds.reset_index("x")
+        with self.roundtrip(ds_reset) as actual:
+            assert_identical(actual, ds_reset)
+
 
 class NetCDFBase(CFEncodedBase):
     """Tests for all netCDF3 and netCDF4 backends."""

From 1b8a636ab3bb56b71946aa58c1fc83978ae5902d Mon Sep 17 00:00:00 2001
From: Benoit Bovy <benbovy@gmail.com>
Date: Fri, 26 Jan 2024 11:49:42 +0100
Subject: [PATCH 3/3] update what's new

---
 doc/whats-new.rst | 3 +++
 1 file changed, 3 insertions(+)

diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index ac0015c14c5..97801d6e6b1 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -37,6 +37,9 @@ Deprecations
 Bug fixes
 ~~~~~~~~~
 
+- Fixed a regression that prevented multi-index level coordinates being
+  serialized after resetting or dropping the multi-index (:issue:`8628`, :pull:`8672`).
+  By `Benoit Bovy <https://github.com/benbovy>`_.
 
 Documentation
 ~~~~~~~~~~~~~