diff --git a/src/Psalm/Internal/Analyzer/ClassAnalyzer.php b/src/Psalm/Internal/Analyzer/ClassAnalyzer.php
index 0ba7100a7..b6a5531d4 100644
--- a/src/Psalm/Internal/Analyzer/ClassAnalyzer.php
+++ b/src/Psalm/Internal/Analyzer/ClassAnalyzer.php
@@ -1585,7 +1585,7 @@ class ClassAnalyzer extends ClassLikeAnalyzer
                 : [];
 
             if ($actual_method_storage->overridden_downstream) {
-                $overridden_method_ids[] = 'overridden::downstream';
+                $overridden_method_ids['overridden::downstream'] = 'overridden::downstream';
             }
 
             if (!$return_type && isset($class_storage->interface_method_ids[strtolower($stmt->name->name)])) {
diff --git a/src/Psalm/Internal/Analyzer/TypeAnalyzer.php b/src/Psalm/Internal/Analyzer/TypeAnalyzer.php
index 631c73d33..bfe2fd3b0 100644
--- a/src/Psalm/Internal/Analyzer/TypeAnalyzer.php
+++ b/src/Psalm/Internal/Analyzer/TypeAnalyzer.php
@@ -264,13 +264,11 @@ class TypeAnalyzer
      *
      * @param  Type\Union   $input_type
      * @param  Type\Union   $container_type
-     *
-     * @return bool
      */
     public static function isSimplyContainedBy(
         Type\Union $input_type,
         Type\Union $container_type
-    ) {
+    ) : bool {
         if ($input_type->getId() === $container_type->getId()) {
             return true;
         }
@@ -285,10 +283,26 @@ class TypeAnalyzer
         $container_type_not_null = clone $container_type;
         $container_type_not_null->removeType('null');
 
-        return !array_diff_key(
-            $input_type_not_null->getTypes(),
-            $container_type_not_null->getTypes()
-        );
+        foreach ($input_type->getTypes() as $input_key => $input_type_part) {
+            foreach ($container_type->getTypes() as $container_key => $container_type_part) {
+                if (get_class($container_type_part) === TNamedObject::class
+                    && $input_type_part instanceof TNamedObject
+                    && $input_type_part->value === $container_type_part->value
+                ) {
+                    continue 2;
+                }
+
+                if ($input_key === $container_key) {
+                    continue 2;
+                }
+            }
+
+            return false;
+        }
+
+
+
+        return true;
     }
 
     /**
diff --git a/src/Psalm/Internal/Codebase/Methods.php b/src/Psalm/Internal/Codebase/Methods.php
index 73e4b2c1f..654b98d7d 100644
--- a/src/Psalm/Internal/Codebase/Methods.php
+++ b/src/Psalm/Internal/Codebase/Methods.php
@@ -554,7 +554,7 @@ class Methods
             $class_storage = $this->classlike_storage_provider->get($fq_class_name);
 
             if ($class_storage->abstract && isset($class_storage->overridden_method_ids[$method_name])) {
-                $appearing_method_id = $class_storage->overridden_method_ids[$method_name][0];
+                $appearing_method_id = reset($class_storage->overridden_method_ids[$method_name]);
             } else {
                 return null;
             }
@@ -757,7 +757,7 @@ class Methods
         }
 
         if ($class_storage->abstract && isset($class_storage->overridden_method_ids[$method_name])) {
-            return $class_storage->overridden_method_ids[$method_name][0];
+            return reset($class_storage->overridden_method_ids[$method_name]);
         }
     }
 
diff --git a/src/Psalm/Internal/Codebase/Populator.php b/src/Psalm/Internal/Codebase/Populator.php
index c058f88e7..b1b89b28a 100644
--- a/src/Psalm/Internal/Codebase/Populator.php
+++ b/src/Psalm/Internal/Codebase/Populator.php
@@ -293,13 +293,13 @@ class Populator
                         && $method_storage->inheritdoc
                         && $declaring_method_storage->throws
                     ) {
-                        $method_storage->throws = $declaring_method_storage->throws;
                     }
 
                     if (count($storage->overridden_method_ids[$method_name]) === 1
                         && $method_storage->signature_return_type
                         && !$method_storage->signature_return_type->isVoid()
-                        && $method_storage->return_type === $method_storage->signature_return_type
+                        && ($method_storage->return_type === $method_storage->signature_return_type
+                            || $method_storage->inherited_return_type)
                     ) {
                         if (isset($declaring_class_storage->methods[$method_name])) {
                             $declaring_method_storage = $declaring_class_storage->methods[$method_name];
@@ -768,7 +768,7 @@ class Populator
                         }
                     }
                 }
-                $storage->overridden_method_ids[$method_name][] = $interface_method_ids[0];
+                $storage->overridden_method_ids[$method_name][$interface_method_ids[0]] = $interface_method_ids[0];
             } else {
                 $storage->interface_method_ids[$method_name] = $interface_method_ids;
             }
@@ -1006,10 +1006,10 @@ class Populator
                     if (isset($declaring_class_storage->methods[$method_name])
                         && $declaring_class_storage->methods[$method_name]->abstract
                     ) {
-                        $storage->overridden_method_ids[$method_name][] = $declaring_method_id;
+                        $storage->overridden_method_ids[$method_name][$declaring_method_id] = $declaring_method_id;
                     }
                 } else {
-                    $storage->overridden_method_ids[$method_name][] = $declaring_method_id;
+                    $storage->overridden_method_ids[$method_name][$declaring_method_id] = $declaring_method_id;
                 }
             }
 
diff --git a/src/Psalm/Internal/Codebase/Reflection.php b/src/Psalm/Internal/Codebase/Reflection.php
index 6613b0ede..10e728165 100644
--- a/src/Psalm/Internal/Codebase/Reflection.php
+++ b/src/Psalm/Internal/Codebase/Reflection.php
@@ -418,7 +418,7 @@ class Reflection
             $storage->declaring_method_ids[$method_name] = $declaring_method_id;
             $storage->inheritable_method_ids[$method_name] = $declaring_method_id;
 
-            $storage->overridden_method_ids[$method_name][] = $declaring_method_id;
+            $storage->overridden_method_ids[$method_name][$declaring_method_id] = $declaring_method_id;
         }
     }
 
diff --git a/src/Psalm/Storage/ClassLikeStorage.php b/src/Psalm/Storage/ClassLikeStorage.php
index 389c4ea0b..2d1390ba2 100644
--- a/src/Psalm/Storage/ClassLikeStorage.php
+++ b/src/Psalm/Storage/ClassLikeStorage.php
@@ -242,7 +242,7 @@ class ClassLikeStorage
     public $appearing_method_ids = [];
 
     /**
-     * @var array<string, array<string>>
+     * @var array<string, array<string, string>>
      */
     public $overridden_method_ids = [];
 
diff --git a/tests/Template/TemplateExtendsTest.php b/tests/Template/TemplateExtendsTest.php
index 6a43f80d5..c2ece1023 100644
--- a/tests/Template/TemplateExtendsTest.php
+++ b/tests/Template/TemplateExtendsTest.php
@@ -1975,6 +1975,21 @@ class TemplateExtendsTest extends TestCase
 
                     echo (new F())->getValue();'
             ],
+            'lessSpecificNonGenericReturnType' => [
+                '<?php
+                    /**
+                     * @template-implements IteratorAggregate<int, int>
+                     */
+                    class Bar implements IteratorAggregate {
+                        public function getIterator() : Traversable {
+                            yield from range(0, 100);
+                        }
+                    }
+
+                    $bat = new Bar();
+
+                    foreach ($bat as $num) {}',
+            ],
         ];
     }