diff --git a/esmvalcore/preprocessor/__init__.py b/esmvalcore/preprocessor/__init__.py
index a57fb50373..90b743b0b0 100644
--- a/esmvalcore/preprocessor/__init__.py
+++ b/esmvalcore/preprocessor/__init__.py
@@ -654,12 +654,8 @@ def _run(self, _):
             for product in self.products for step in product.settings
         }
         blocks = get_step_blocks(steps, self.order)
-        if not blocks:
-            # If no preprocessing is configured, just load the data and save.
-            for product in self.products:
-                product.cubes  # pylint: disable=pointless-statement
-                product.close()
 
+        saved = set()
         for block in blocks:
             logger.debug("Running block %s", block)
             if block[0] in MULTI_MODEL_FUNCTIONS:
@@ -675,9 +671,13 @@ def _run(self, _):
                     if block == blocks[-1]:
                         product.cubes  # pylint: disable=pointless-statement
                         product.close()
+                        saved.add(product.filename)
 
         for product in self.products:
-            product.close()
+            if product.filename not in saved:
+                product.cubes  # pylint: disable=pointless-statement
+                product.close()
+
         metadata_files = write_metadata(self.products,
                                         self.write_ncl_interface)
         return metadata_files
diff --git a/tests/integration/preprocessor/test_preprocessing_task.py b/tests/integration/preprocessor/test_preprocessing_task.py
index 99d971cf8c..81f980c9e8 100644
--- a/tests/integration/preprocessor/test_preprocessing_task.py
+++ b/tests/integration/preprocessor/test_preprocessing_task.py
@@ -3,6 +3,7 @@
 import iris.cube
 from prov.model import ProvDocument
 
+import esmvalcore.preprocessor
 from esmvalcore.dataset import Dataset
 from esmvalcore.preprocessor import PreprocessingTask, PreprocessorFile
 
@@ -15,7 +16,7 @@ def test_load_save_task(tmp_path):
     iris.save(cube, in_file)
     dataset = Dataset(short_name='tas')
     dataset.files = [in_file]
-    dataset._load_with_callback = lambda _: cube
+    dataset._load_with_callback = lambda _: cube.copy()
 
     # Create task
     task = PreprocessingTask([
@@ -40,3 +41,108 @@ def test_load_save_task(tmp_path):
 
     result.attributes.clear()
     assert result == cube
+
+
+def test_load_save_and_other_task(tmp_path, monkeypatch):
+    """Test that a task just copies one file and preprocesses another file."""
+    # Prepare test datasets
+    in_cube = iris.cube.Cube(data=[0.], var_name='tas', units='degrees_C')
+    (tmp_path / 'climate_data').mkdir()
+    file1 = tmp_path / 'climate_data' / 'tas_dataset1.nc'
+    file2 = tmp_path / 'climate_data' / 'tas_dataset2.nc'
+
+    # Save cubes for reading global attributes into provenance
+    iris.save(in_cube, target=file1)
+    iris.save(in_cube, target=file2)
+
+    dataset1 = Dataset(short_name='tas', dataset='dataset1')
+    dataset1.files = [file1]
+    dataset1._load_with_callback = lambda _: in_cube.copy()
+
+    dataset2 = Dataset(short_name='tas', dataset='dataset1')
+    dataset2.files = [file2]
+    dataset2._load_with_callback = lambda _: in_cube.copy()
+
+    # Create some mock preprocessor functions and patch
+    # `esmvalcore.preprocessor` so it uses them.
+    def single_preproc_func(cube):
+        cube.data = cube.core_data() + 1.
+        return cube
+
+    def multi_preproc_func(products):
+        for product in products:
+            cube = product.cubes[0]
+            cube.data = cube.core_data() + 1.
+            product.cubes = [cube]
+        return products
+
+    monkeypatch.setattr(
+        esmvalcore.preprocessor,
+        'single_preproc_func',
+        single_preproc_func,
+        raising=False,
+    )
+    monkeypatch.setattr(
+        esmvalcore.preprocessor,
+        'multi_preproc_func',
+        multi_preproc_func,
+        raising=False,
+    )
+    monkeypatch.setattr(
+        esmvalcore.preprocessor,
+        'MULTI_MODEL_FUNCTIONS',
+        {'multi_preproc_func'},
+    )
+    default_order = (esmvalcore.preprocessor.INITIAL_STEPS +
+                     ('single_preproc_func', 'multi_preproc_func') +
+                     esmvalcore.preprocessor.FINAL_STEPS)
+    monkeypatch.setattr(
+        esmvalcore.preprocessor,
+        'DEFAULT_ORDER',
+        default_order,
+    )
+
+    # Create task
+    task = PreprocessingTask(
+        [
+            PreprocessorFile(
+                filename=tmp_path / 'tas_dataset1.nc',
+                settings={},
+                datasets=[dataset1],
+                attributes={'dataset': 'dataset1'},
+            ),
+            PreprocessorFile(
+                filename=tmp_path / 'tas_dataset2.nc',
+                settings={
+                    'single_preproc_func': {},
+                    'multi_preproc_func': {},
+                },
+                datasets=[dataset2],
+                attributes={'dataset': 'dataset2'},
+            ),
+        ],
+        order=default_order,
+    )
+
+    # Create an 'activity' representing a run of the tool
+    provenance = ProvDocument()
+    provenance.add_namespace('software', uri='https://example.com/software')
+    activity = provenance.activity('software:esmvalcore')
+    task.initialize_provenance(activity)
+
+    task.run()
+
+    # Check that two files were saved and the preprocessor functions were
+    # only applied to the second one.
+    assert len(task.products) == 2
+    for product in task.products:
+        print(product.filename)
+        assert product.filename.exists()
+        out_cube = iris.load_cube(product.filename)
+        print(out_cube.data)
+        if product.attributes['dataset'] == 'dataset1':
+            assert out_cube.data.tolist() == [0.]
+        elif product.attributes['dataset'] == 'dataset2':
+            assert out_cube.data.tolist() == [2.]
+        else:
+            assert False, "unexpected product"