diff --git a/docs/src/conf.py b/docs/src/conf.py
index 312925b4ea..94106b952f 100644
--- a/docs/src/conf.py
+++ b/docs/src/conf.py
@@ -382,6 +382,7 @@ def _dotv(version):
     "http://www.esrl.noaa.gov/psd/data/gridded/conventions/cdc_netcdf_standard.shtml",
     "http://www.nationalarchives.gov.uk/doc/open-government-licence",
     "https://www.metoffice.gov.uk/",
+    "https://biggus.readthedocs.io/",
 ]
 
 # list of sources to exclude from the build.
diff --git a/docs/src/whatsnew/3.6.rst b/docs/src/whatsnew/3.6.rst
index 151c63ef51..543ce7372e 100644
--- a/docs/src/whatsnew/3.6.rst
+++ b/docs/src/whatsnew/3.6.rst
@@ -55,6 +55,23 @@ This document explains the changes made to Iris for this release
     or feature requests for improving Iris. Enjoy!
 
 
+|iris_version| |build_date|
+===========================
+
+.. dropdown:: |iris_version| Patches
+   :color: primary
+   :icon: alert
+   :animate: fade-in
+
+   The patches in this release of Iris include:
+
+   🐛 **Bugs Fixed**
+
+   #. `@stephenworsley`_ fixed :meth:`~iris.cube.Cube.convert_units` to allow unit
+      conversion of lazy data when using a `Distributed`_ scheduler.
+      (:issue:`5347`, :pull:`5349`)
+
+
 📢 Announcements
 ================
 
@@ -180,4 +197,5 @@ This document explains the changes made to Iris for this release
 .. _PEP-0621: https://peps.python.org/pep-0621/
 .. _pypa/build: https://pypa-build.readthedocs.io/en/stable/
 .. _NEP29: https://numpy.org/neps/nep-0029-deprecation_policy.html
-.. _Contributor Covenant: https://www.contributor-covenant.org/version/2/1/code_of_conduct/
\ No newline at end of file
+.. _Contributor Covenant: https://www.contributor-covenant.org/version/2/1/code_of_conduct/
+.. _Distributed: https://distributed.dask.org/en/stable/
diff --git a/lib/iris/cube.py b/lib/iris/cube.py
index 7c6fd55c10..44a40cf72b 100644
--- a/lib/iris/cube.py
+++ b/lib/iris/cube.py
@@ -19,6 +19,7 @@
 from xml.dom.minidom import Document
 import zlib
 
+from cf_units import Unit
 import dask.array as da
 import numpy as np
 import numpy.ma as ma
@@ -1144,7 +1145,7 @@ def convert_units(self, unit):
             )
         if self.has_lazy_data():
             # Make fixed copies of old + new units for a delayed conversion.
-            old_unit = self.units
+            old_unit = Unit(self.units)
             new_unit = unit
 
             pointwise_convert = partial(old_unit.convert, other=new_unit)
diff --git a/lib/iris/tests/unit/cube/test_Cube.py b/lib/iris/tests/unit/cube/test_Cube.py
index aa9e3b51b1..28fbe429c1 100644
--- a/lib/iris/tests/unit/cube/test_Cube.py
+++ b/lib/iris/tests/unit/cube/test_Cube.py
@@ -14,6 +14,8 @@
 from unittest import mock
 
 from cf_units import Unit
+import dask.array as da
+from distributed import Client
 import numpy as np
 import numpy.ma as ma
 import pytest
@@ -3012,6 +3014,14 @@ def test_preserves_lazy(self):
         self.assertTrue(cube.has_lazy_data())
         self.assertArrayAllClose(cube.data, real_data_ft)
 
+    def test_unit_multiply(self):
+        _client = Client()
+        cube = iris.cube.Cube(da.arange(1), units="m")
+        cube.units *= "s-1"
+        cube.convert_units("m s-1")
+        cube.data
+        _client.close()
+
 
 class Test__eq__data(tests.IrisTest):
     """Partial cube equality testing, for data type only."""