diff --git a/packaging/_parser.py b/packaging/_parser.py
index 378cef88..29cb27d3 100644
--- a/packaging/_parser.py
+++ b/packaging/_parser.py
@@ -120,7 +120,13 @@ def _parse_requirement_details(
             return (url, specifier, marker)
 
         marker = _parse_requirement_marker(
-            tokenizer, span_start=specifier_start, after="version specifier"
+            tokenizer,
+            span_start=specifier_start,
+            after=(
+                "version specifier"
+                if specifier
+                else "name and no valid version specifier"
+            ),
         )
 
     return (url, specifier, marker)
diff --git a/tests/test_requirements.py b/tests/test_requirements.py
index 09c963d0..5305fbdf 100644
--- a/tests/test_requirements.py
+++ b/tests/test_requirements.py
@@ -489,11 +489,27 @@ def test_error_on_missing_op_after_name(self) -> None:
         # THEN
         assert ctx.exconly() == (
             "packaging.requirements.InvalidRequirement: "
-            "Expected end or semicolon (after version specifier)\n"
+            "Expected end or semicolon (after name and no valid version specifier)\n"
             "    name 1.0\n"
             "         ^"
         )
 
+    def test_error_on_random_char_after_specifier(self) -> None:
+        # GIVEN
+        to_parse = "name >= 1.0 #"
+
+        # WHEN
+        with pytest.raises(InvalidRequirement) as ctx:
+            Requirement(to_parse)
+
+        # THEN
+        assert ctx.exconly() == (
+            "packaging.requirements.InvalidRequirement: "
+            "Expected end or semicolon (after version specifier)\n"
+            "    name >= 1.0 #\n"
+            "         ~~~~~~~^"
+        )
+
 
 class TestRequirementBehaviour:
     def test_types_with_nothing(self) -> None: