diff --git a/docs/src/whatsnew/latest.rst b/docs/src/whatsnew/latest.rst
index 35359c4482..776e1a0217 100644
--- a/docs/src/whatsnew/latest.rst
+++ b/docs/src/whatsnew/latest.rst
@@ -96,6 +96,10 @@ This document explains the changes made to Iris for this release
    constraints are given. This was previously only implemented where one such
    constraint was given. (:issue:`6228`, :pull:`6754`)
 
+#. `@stephenworsley`_ reduced the memory load for regridding and other operations
+   using :func:`~iris._lazy_data.map_complete_blocks` when the output chunks would
+   exceed the optimum chunksize set in dask. (:pull:`6730`)
+
 
 🔥 Deprecations
 ===============
diff --git a/lib/iris/_lazy_data.py b/lib/iris/_lazy_data.py
index 992789ed90..c74da78d6b 100644
--- a/lib/iris/_lazy_data.py
+++ b/lib/iris/_lazy_data.py
@@ -626,6 +626,14 @@ def map_complete_blocks(src, func, dims, out_sizes, dtype, *args, **kwargs):
     --------
     :func:`dask.array.map_blocks` : The function used for the mapping.
 
+    Notes
+    -----
+    .. note:
+
+        If the output chunks would be larger than the maximum chunksize set
+        in the dask config, the input is rechunked, where possible to
+        optimise the output chunksize.
+
     """
     data = None
     result = None
@@ -640,17 +648,40 @@ def map_complete_blocks(src, func, dims, out_sizes, dtype, *args, **kwargs):
     else:
         data = src.lazy_data()
 
+    shape = list(src.shape)
+
     if result is None and data is not None:
         # Ensure dims are not chunked
         in_chunks = list(data.chunks)
         for dim in dims:
-            in_chunks[dim] = src.shape[dim]
-        data = data.rechunk(in_chunks)
+            in_chunks[dim] = (src.shape[dim],)
 
         # Determine output chunks
-        out_chunks = list(data.chunks)
+        out_chunks = in_chunks.copy()
         for dim, size in zip(dims, out_sizes):
-            out_chunks[dim] = size
+            out_chunks[dim] = (size,)
+            shape[dim] = size
+
+        # Ensure the chunksize of the output is a reasonable size.
+        max_outchunks = [max(chunk) for chunk in out_chunks]
+        df = tuple(i in dims for i in range(len(shape)))
+        dtype = np.dtype(dtype)
+        opt_outchunks = _optimum_chunksize(
+            max_outchunks, shape, dtype=dtype, dims_fixed=df
+        )
+        for i, (chunk, max_out, opt_out) in enumerate(
+            zip(out_chunks, max_outchunks, opt_outchunks)
+        ):
+            if opt_out < max_out:
+                new_chunks = []
+                for c in chunk:
+                    new_chunks.extend((c // opt_out) * [opt_out])
+                    if chunk_end := c % opt_out:
+                        new_chunks.append(chunk_end)
+                in_chunks[i] = tuple(new_chunks)
+                out_chunks[i] = tuple(new_chunks)
+
+        data = data.rechunk(in_chunks)
 
         # Assume operation preserves mask.
         meta = da.utils.meta_from_array(data).astype(dtype)
diff --git a/lib/iris/tests/unit/lazy_data/test_map_complete_blocks.py b/lib/iris/tests/unit/lazy_data/test_map_complete_blocks.py
index be25ab6c09..6dc8286996 100644
--- a/lib/iris/tests/unit/lazy_data/test_map_complete_blocks.py
+++ b/lib/iris/tests/unit/lazy_data/test_map_complete_blocks.py
@@ -7,6 +7,7 @@
 from unittest.mock import Mock, PropertyMock
 
 import dask.array as da
+import dask.config
 import numpy as np
 
 from iris._lazy_data import is_lazy_data, map_complete_blocks
@@ -134,3 +135,21 @@ def test_multidimensional_input(self):
         )
         assert is_lazy_data(result)
         assert_array_equal(result.compute(), array + 1)
+
+    def test_rechunking(self):
+        # Choose a dask array with an irregularly chunked dimension to be rechunked.
+        lazy_array = da.ones((5, 10, 9, 10), chunks=(2, 10, 5, 5))
+        cube, _ = create_mock_cube(lazy_array)
+
+        # Reduce the optimum dask chunksize.
+        with dask.config.set({"array.chunk-size": "32KiB"}):
+            result = map_complete_blocks(
+                cube, self.func, dims=(1, 3), out_sizes=(30, 40), dtype=lazy_array.dtype
+            )
+            assert is_lazy_data(result)
+            expected_chunksize = (1, 30, 2, 40)
+            assert result.chunksize == expected_chunksize
+            # Note that one chunk is irregularly rechunked and the other isn't.
+            assert result.chunks[0] == (1, 1, 1, 1, 1)
+            # split from the original chunks of (5, 4)
+            assert result.chunks[2] == (2, 2, 1, 2, 2)