diff --git a/xarray/core/indexing.py b/xarray/core/indexing.py
index 08b1d0be290..d8727c38c48 100644
--- a/xarray/core/indexing.py
+++ b/xarray/core/indexing.py
@@ -878,10 +878,10 @@ def as_indexable(array):
         return PandasIndexingAdapter(array)
     if is_duck_dask_array(array):
         return DaskIndexingAdapter(array)
-    if hasattr(array, "__array_function__"):
-        return NdArrayLikeIndexingAdapter(array)
     if hasattr(array, "__array_namespace__"):
         return ArrayApiIndexingAdapter(array)
+    if hasattr(array, "__array_function__"):
+        return NdArrayLikeIndexingAdapter(array)
 
     raise TypeError(f"Invalid array type: {type(array)}")
 
diff --git a/xarray/core/nanops.py b/xarray/core/nanops.py
index fc7240139aa..7fbb63068c0 100644
--- a/xarray/core/nanops.py
+++ b/xarray/core/nanops.py
@@ -162,7 +162,7 @@ def nanstd(a, axis=None, dtype=None, out=None, ddof=0):
 
 def nanprod(a, axis=None, dtype=None, out=None, min_count=None):
     mask = isnull(a)
-    result = nputils.nanprod(a, axis=axis, dtype=dtype, out=out)
+    result = nputils.nanprod(a, axis=axis, dtype=dtype)
     if min_count is not None:
         return _maybe_null_out(result, axis, mask, min_count)
     else:
diff --git a/xarray/tests/test_indexing.py b/xarray/tests/test_indexing.py
index 985fb02a87e..92c21cc32fb 100644
--- a/xarray/tests/test_indexing.py
+++ b/xarray/tests/test_indexing.py
@@ -894,6 +894,74 @@ def test_posify_mask_subindexer(indices, expected) -> None:
     np.testing.assert_array_equal(expected, actual)
 
 
+class ArrayWithNamespace:
+    def __array_namespace__(self, version=None):
+        pass
+
+
+class ArrayWithArrayFunction:
+    def __array_function__(self, func, types, args, kwargs):
+        pass
+
+
+class ArrayWithNamespaceAndArrayFunction:
+    def __array_namespace__(self, version=None):
+        pass
+
+    def __array_function__(self, func, types, args, kwargs):
+        pass
+
+
+def as_dask_array(arr, chunks):
+    try:
+        import dask.array as da
+    except ImportError:
+        return None
+
+    return da.from_array(arr, chunks=chunks)
+
+
+@pytest.mark.parametrize(
+    ["array", "expected_type"],
+    (
+        pytest.param(
+            indexing.CopyOnWriteArray(np.array([1, 2])),
+            indexing.CopyOnWriteArray,
+            id="ExplicitlyIndexed",
+        ),
+        pytest.param(
+            np.array([1, 2]), indexing.NumpyIndexingAdapter, id="numpy.ndarray"
+        ),
+        pytest.param(
+            pd.Index([1, 2]), indexing.PandasIndexingAdapter, id="pandas.Index"
+        ),
+        pytest.param(
+            as_dask_array(np.array([1, 2]), chunks=(1,)),
+            indexing.DaskIndexingAdapter,
+            id="dask.array",
+            marks=requires_dask,
+        ),
+        pytest.param(
+            ArrayWithNamespace(), indexing.ArrayApiIndexingAdapter, id="array_api"
+        ),
+        pytest.param(
+            ArrayWithArrayFunction(),
+            indexing.NdArrayLikeIndexingAdapter,
+            id="array_like",
+        ),
+        pytest.param(
+            ArrayWithNamespaceAndArrayFunction(),
+            indexing.ArrayApiIndexingAdapter,
+            id="array_api_with_fallback",
+        ),
+    ),
+)
+def test_as_indexable(array, expected_type):
+    actual = indexing.as_indexable(array)
+
+    assert isinstance(actual, expected_type)
+
+
 def test_indexing_1d_object_array() -> None:
     items = (np.arange(3), np.arange(6))
     arr = DataArray(np.array(items, dtype=object))