diff --git a/xarray/core/variable.py b/xarray/core/variable.py
index 9271d0c4dbd..7c84f6adb49 100644
--- a/xarray/core/variable.py
+++ b/xarray/core/variable.py
@@ -161,12 +161,14 @@ def as_variable(obj, name=None) -> Variable | IndexVariable:
     if name is not None and name in obj.dims:
         # convert the Variable into an Index
         if obj.ndim != 1:
-            raise MissingDimensionsError(
+            warnings.warn(
                 f"{name!r} has more than 1-dimension and the same name as one of its "
-                f"dimensions {obj.dims!r}. xarray disallows such variables because they "
-                "conflict with the coordinates used to label dimensions."
+                f"dimensions {obj.dims!r}. Xarray will not automatically"
+                "create an Index object that would allow label-based selection.",
+                RuntimeWarning,
             )
-        obj = obj.to_index_variable()
+        else:
+            obj = obj.to_index_variable()
 
     return obj