diff --git a/cosmos/operators/local.py b/cosmos/operators/local.py
index 4888583bb..36c676017 100644
--- a/cosmos/operators/local.py
+++ b/cosmos/operators/local.py
@@ -40,9 +40,7 @@
     FullOutputSubprocessHook,
     FullOutputSubprocessResult,
 )
-from cosmos.dbt.parser.output import (
-    extract_log_issues,
-)
+from cosmos.dbt.parser.output import extract_log_issues, parse_output
 
 logger = get_logger(__name__)
 
@@ -350,11 +348,12 @@ def get_openlineage_facets_on_complete(self, task_instance: TaskInstance) -> Ope
             job_facets=job_facets,
         )
 
-    def build_and_run_cmd(self, context: Context, cmd_flags: list[str] | None = None) -> None:
+    def build_and_run_cmd(self, context: Context, cmd_flags: list[str] | None = None) -> FullOutputSubprocessResult:
         dbt_cmd, env = self.build_cmd(context=context, cmd_flags=cmd_flags)
         dbt_cmd = dbt_cmd or []
         result = self.run_command(cmd=dbt_cmd, env=env, context=context)
         logger.info(result.output)
+        return result
 
     def execute(self, context: Context) -> None:
         self.build_and_run_cmd(context=context)
@@ -472,7 +471,6 @@ def _should_run_tests(
 
         :param result: The output from the build and run command.
         """
-
         return self.on_warning_callback is not None and no_tests_message not in result.output
 
     def _handle_warnings(self, result: FullOutputSubprocessResult, context: Context) -> None:
@@ -492,6 +490,13 @@ def _handle_warnings(self, result: FullOutputSubprocessResult, context: Context)
         if self.on_warning_callback:
             self.on_warning_callback(warning_context)
 
+    def execute(self, context: Context) -> None:
+        result = self.build_and_run_cmd(context=context)
+        if self._should_run_tests(result):
+            warnings = parse_output(result, "WARN")
+            if warnings > 0:
+                self._handle_warnings(result, context)
+
 
 class DbtRunOperationLocalOperator(DbtLocalBaseOperator):
     """
diff --git a/tests/operators/test_local.py b/tests/operators/test_local.py
index 94b0f8e27..223305fbe 100644
--- a/tests/operators/test_local.py
+++ b/tests/operators/test_local.py
@@ -1,3 +1,6 @@
+import os
+import shutil
+import tempfile
 from pathlib import Path
 from unittest.mock import MagicMock, patch
 
@@ -28,6 +31,8 @@
 
 
 DBT_PROJ_DIR = Path(__file__).parent.parent.parent / "dev/dags/dbt/jaffle_shop"
+SCHEMA_FAILING_TEST = Path(__file__).parent.parent / "sample/schema_failing_test.yml"
+
 
 profile_config = ProfileConfig(
     profile_name="default",
@@ -45,6 +50,18 @@
 )
 
 
+@pytest.fixture
+def failing_test_dbt_project(tmp_path):
+    tmp_dir = tempfile.TemporaryDirectory()
+    tmp_dir_path = Path(tmp_dir.name) / "jaffle_shop"
+    shutil.copytree(DBT_PROJ_DIR, tmp_dir_path)
+    target_schema = tmp_dir_path / "models/schema.yml"
+    os.remove(target_schema)
+    shutil.copy(SCHEMA_FAILING_TEST, target_schema)
+    yield tmp_dir_path
+    tmp_dir.cleanup()
+
+
 def test_dbt_base_operator_add_global_flags() -> None:
     dbt_base_operator = DbtLocalBaseOperator(
         profile_config=profile_config,
@@ -175,7 +192,7 @@ def test_run_operator_dataset_inlets_and_outlets():
             dbt_cmd_flags=["--models", "stg_customers"],
             install_deps=True,
         )
-        run_operator
+        run_operator >> test_operator
     run_test_dag(dag)
     assert run_operator.inlets == []
     assert run_operator.outlets == [Dataset(uri="postgres://0.0.0.0:5432/postgres.public.stg_customers", extra=None)]
@@ -183,6 +200,31 @@ def test_run_operator_dataset_inlets_and_outlets():
     assert test_operator.outlets == []
 
 
+@pytest.mark.integration
+def test_run_test_operator_with_callback(failing_test_dbt_project):
+    on_warning_callback = MagicMock()
+
+    with DAG("test-id-2", start_date=datetime(2022, 1, 1)) as dag:
+        run_operator = DbtRunLocalOperator(
+            profile_config=real_profile_config,
+            project_dir=failing_test_dbt_project,
+            task_id="run",
+            dbt_cmd_flags=["--models", "orders"],
+            install_deps=True,
+        )
+        test_operator = DbtTestLocalOperator(
+            profile_config=real_profile_config,
+            project_dir=failing_test_dbt_project,
+            task_id="test",
+            dbt_cmd_flags=["--models", "orders"],
+            install_deps=True,
+            on_warning_callback=on_warning_callback,
+        )
+        run_operator >> test_operator
+    run_test_dag(dag)
+    assert on_warning_callback.called
+
+
 @pytest.mark.integration
 def test_run_operator_emits_events():
     class MockRun:
diff --git a/tests/sample/schema_failing_test.yml b/tests/sample/schema_failing_test.yml
new file mode 100644
index 000000000..c75df8152
--- /dev/null
+++ b/tests/sample/schema_failing_test.yml
@@ -0,0 +1,83 @@
+version: 2
+
+models:
+  - name: customers
+    description: This table has basic information about a customer, as well as some derived facts based on a customer's orders
+
+    columns:
+      - name: customer_id
+        description: This is a unique identifier for a customer
+        tests:
+          - unique
+          - not_null
+
+      - name: first_name
+        description: Customer's first name. PII.
+
+      - name: last_name
+        description: Customer's last name. PII.
+
+      - name: first_order
+        description: Date (UTC) of a customer's first order
+
+      - name: most_recent_order
+        description: Date (UTC) of a customer's most recent order
+
+      - name: number_of_orders
+        description: Count of the number of orders a customer has placed
+
+      - name: total_order_amount
+        description: Total value (AUD) of a customer's orders
+
+  - name: orders
+    description: This table has basic information about orders, as well as some derived facts based on payments
+
+    columns:
+      - name: order_id
+        tests:
+          - unique
+          - not_null
+        description: This is a unique identifier for an order
+
+      - name: customer_id
+        description: Foreign key to the customers table
+        tests:
+          - not_null
+          - relationships:
+              to: ref('customers')
+              field: customer_id
+
+      - name: order_date
+        description: Date (UTC) that the order was placed
+
+      - name: status
+        description: '{{ doc("orders_status") }}'
+        tests:
+          - accepted_values:
+              # this test will fail, since this column has more values
+              values: ['placed']
+
+      - name: amount
+        description: Total amount (AUD) of the order
+        tests:
+          - not_null
+
+      - name: credit_card_amount
+        description: Amount of the order (AUD) paid for by credit card
+        tests:
+          - not_null
+
+      - name: coupon_amount
+        description: Amount of the order (AUD) paid for by coupon
+        tests:
+          - not_null
+
+      - name: bank_transfer_amount
+        description: Amount of the order (AUD) paid for by bank transfer
+        tests:
+          - not_null
+
+      - name: gift_card_amount
+        description: Amount of the order (AUD) paid for by gift card
+        tests:
+          - not_null