diff --git a/asdf/_tests/test_extension.py b/asdf/_tests/test_extension.py
index 6adc39a6b..59b37e311 100644
--- a/asdf/_tests/test_extension.py
+++ b/asdf/_tests/test_extension.py
@@ -1,5 +1,6 @@
 import pytest
 from packaging.specifiers import SpecifierSet
+from yaml.representer import RepresenterError
 
 from asdf import AsdfFile, config_context
 from asdf._tests._helpers import assert_extension_correctness
@@ -607,6 +608,38 @@ def test_converter_proxy():
         ConverterProxy(MinimumConverter(tags=[extension.tags[0].tag_uri], types=[object()]), extension)
 
 
+def test_converter_subclass_with_no_supported_tags():
+    """
+    Adding a Converter to an Extension that doesn't list support for the tags
+    associated with the Converter should not index the types listed for the
+    Converter.
+    """
+
+    class Foo:
+        pass
+
+    class FooConverterWithSubclass(Converter):
+        tags = ["asdf://somewhere.org/tags/foo-1.0.0"]
+        types = [Foo]
+
+        def to_yaml_tree(self, *args):
+            pass
+
+        def from_yaml_tree(self, *args):
+            pass
+
+    class FooExtension(Extension):
+        tags = []
+        converters = [FooConverterWithSubclass()]
+        extension_uri = "asdf://somewhere.org/extensions/foo-1.0.0"
+
+    tree = {"obj": Foo()}
+    with config_context() as cfg:
+        cfg.add_extension(FooExtension())
+        with pytest.raises(RepresenterError, match=r"cannot represent an object"):
+            roundtrip_object(tree)
+
+
 def test_get_cached_asdf_extension_list():
     extension = LegacyExtension()
     extension_list = get_cached_asdf_extension_list([extension])
diff --git a/asdf/extension/_converter.py b/asdf/extension/_converter.py
index 4044f9d5c..80a3ea341 100644
--- a/asdf/extension/_converter.py
+++ b/asdf/extension/_converter.py
@@ -3,6 +3,7 @@
 types.  Will eventually replace the `asdf.types` module.
 """
 import abc
+import inspect
 
 from asdf.util import get_class_name, uri_match
 
@@ -211,9 +212,25 @@ def __init__(self, delegate, extension):
 
         self._types = []
 
-        if not len(self._tags) and not hasattr(delegate, "select_tag"):
-            # this converter supports no tags so don't inspect the types
-            return
+        if not len(self._tags):
+            # this converter supports no tags supported by the extension
+            # before indexing it's types we need to check select_tag
+
+            # check if it implements select_tag (and not just because it
+            # subclasses Converter)
+            if not hasattr(delegate, "select_tag"):
+                return
+
+            # find which class implements select_tag
+            for class_ in inspect.getmro(delegate.__class__):
+                if "select_tag" in class_.__dict__:
+                    if class_ is not Converter:
+                        # a non-Converter class implements select_tag so consider the types for this Converter
+                        break
+                    else:
+                        # this converter implements select_tag only because it inherits from Converter
+                        # don't index it's types
+                        return
 
         for typ in delegate.types:
             if isinstance(typ, (str, type)):