diff --git a/doc/sphinx/source/tutorials/hessian2mc.rst b/doc/sphinx/source/tutorials/hessian2mc.rst
new file mode 100644
index 0000000000..344bece3ab
--- /dev/null
+++ b/doc/sphinx/source/tutorials/hessian2mc.rst
@@ -0,0 +1,33 @@
+.. _hessian2mc:
+How to transform a Hessian set into the corresponding Monte Carlo PDF
+=====================================================================
+
+A Hessian PDF set can be transformed into a Monte Carlo replica set using the
+method described in Eq. (4.3) of `PDF4LHC21 <https://arxiv.org/pdf/2203.05506>`_ using a runcard 
+like the one below. 
+In this example ``Neig`` are the number of basis eigenvectors of the Hessian PDF. 
+``mc_pdf_name`` is the name of the new MC replica set that will be created. 
+``watt_thorne_rnd_seed`` is the random seed used to generate the MC replicas as shown in the equation below.
+
+.. math::
+
+    f^{MC, (k)}(x,Q) = f^{H}_{0}(x,Q) + \frac{1}{2} \sum_{j=1}^{Neig} \left(f^{H}_{j,+}(x,Q) - f^{H}_{j,-}(x,Q)\right) R_j^{(k)}
+
+where :math:`f^{MC, (k)}(x,Q)` is the :math:`k`-th Monte Carlo replica, :math:`f^{H}_{0}(x,Q)` the central Hessian member,
+:math:`f^{H}_{j,\pm}(x,Q)` the positive and negative eigenvector directions corresponding to the :math:`j`-th Hessian eigenvector, and :math:`R_j^{(k)}`
+are random standard normal numbers.
+
+The new MC replica set will be saved in the LHAPDF folder.
+
+.. code:: yaml
+
+    pdf: MSTW2008nlo68cl 
+
+    mc_pdf_name: MSTW2008nlo68cl_mc
+
+    num_members: 100
+
+    watt_thorne_rnd_seed: 1
+
+    actions_:
+    - write_hessian_to_mc_watt_thorne
diff --git a/doc/sphinx/source/tutorials/index.rst b/doc/sphinx/source/tutorials/index.rst
index a9902db814..c6c20abf63 100644
--- a/doc/sphinx/source/tutorials/index.rst
+++ b/doc/sphinx/source/tutorials/index.rst
@@ -47,6 +47,7 @@ Special PDF sets
    ./reproduce
    ./bundle-pdfs.rst
    ./mc2hessian.rst
+   ./hessian2mc.rst
 
 Miscellaneous
 -------------
diff --git a/validphys2/examples/hessian_2mc.yaml b/validphys2/examples/hessian_2mc.yaml
new file mode 100644
index 0000000000..6a3cf4c895
--- /dev/null
+++ b/validphys2/examples/hessian_2mc.yaml
@@ -0,0 +1,16 @@
+meta:
+  author: Lazy Person
+  title: runcard for conversion of msht20 hessian set to MC
+  keywords: [hessian Monte carlo]
+
+
+pdf: MSTW2008nlo68cl
+
+mc_pdf_name: MSTW2008nlo68cl_mc
+
+num_members: 100
+
+watt_thorne_rnd_seed: 1
+
+actions_:
+  - write_hessian_to_mc_watt_thorne
diff --git a/validphys2/src/validphys/app.py b/validphys2/src/validphys/app.py
index 8073566389..42cf0def31 100644
--- a/validphys2/src/validphys/app.py
+++ b/validphys2/src/validphys/app.py
@@ -52,6 +52,7 @@
     "validphys.mc2hessian",
     "reportengine.report",
     "validphys.overfit_metric",
+    "validphys.hessian2mc",
 ]
 
 log = logging.getLogger(__name__)
diff --git a/validphys2/src/validphys/checks.py b/validphys2/src/validphys/checks.py
index 8ccecdec43..645c37bb88 100644
--- a/validphys2/src/validphys/checks.py
+++ b/validphys2/src/validphys/checks.py
@@ -42,6 +42,15 @@ def check_pdf_is_montecarlo_or_hessian(pdf, **kwargs):
     )
 
 
+@make_argcheck
+def check_pdf_is_hessian(pdf, **kwargs):
+    etype = pdf.error_type
+    check(
+        etype in {'symmhessian', 'hessian'},
+        f"Error type of PDF {pdf} must be 'symmhessian' or 'hessian' and not '{etype}'",
+    )
+
+
 @make_argcheck
 def check_using_theory_covmat(use_theorycovmat):
     """Check that the `use_theorycovmat` is set to True"""
diff --git a/validphys2/src/validphys/hessian2mc.py b/validphys2/src/validphys/hessian2mc.py
new file mode 100644
index 0000000000..c9361c7156
--- /dev/null
+++ b/validphys2/src/validphys/hessian2mc.py
@@ -0,0 +1,145 @@
+"""
+validphys.hessian2mc.py
+
+This module contains the functions that can be used to convert Hessian sets
+like MSHT20 and CT18 to Monte Carlo sets.
+The functions implemented here follow equations (4.3) of the paper arXiv:2203.05506
+"""
+
+import pathlib
+import lhapdf
+import os
+import logging
+import numpy as np
+
+from validphys.lhio import load_all_replicas, rep_matrix, write_replica
+from validphys.checks import check_pdf_is_hessian
+
+log = logging.getLogger(__name__)
+
+
+def write_new_lhapdf_info_file_from_previous_pdf(
+    path_old_pdfset,
+    name_old_pdfset,
+    path_new_pdfset,
+    name_new_pdfset,
+    num_members,
+    description_set="MC representation of hessian PDF set",
+    errortype="replicas",
+):
+    """
+    Writes a new LHAPDF set info file based on an existing set.
+    """
+
+    # write LHAPDF info file for a new pdf set
+    with open(path_old_pdfset / f"{name_old_pdfset}.info", "r") as in_stream, open(
+        path_new_pdfset / f"{name_new_pdfset}.info", "w"
+    ) as out_stream:
+        for l in in_stream.readlines():
+            if l.find("SetDesc:") >= 0:
+                out_stream.write(f'SetDesc: "{description_set}"\n')
+            elif l.find("NumMembers:") >= 0:
+                out_stream.write(f"NumMembers: {num_members+1}\n")
+            elif l.find("ErrorType:") >= 0:
+                out_stream.write(f"ErrorType: {errortype}\n")
+            elif l.find("ErrorConfLevel") >= 0:
+                # remove ErrorConfLevel line
+                pass
+            else:
+                out_stream.write(l)
+    log.info(f"Info file written to {path_new_pdfset / f'{name_new_pdfset}.info'}")
+
+
+def write_mc_watt_thorne_replicas(Rjk_std_normal, replicas_df, mc_pdf_path):
+    """
+    Writes the Monte Carlo representation of a PDF set that is in Hessian form
+    using the Watt-Thorne prescription described in Eq. 4.3 of arXiv:2203.05506.
+
+    Parameters
+    ----------
+    Rjk_std_normal: np.ndarray
+        Array of shape (num_members, n_eig) containing random standard normal numbers.
+
+    replicas_df: pd.DataFrame
+        DataFrame containing replicas of the hessian set at all scales.
+
+    mc_pdf_path: pathlib.Path
+        Path to the new Monte Carlo PDF set.
+    """
+
+    for i, rnd_std_norm_vec in enumerate(Rjk_std_normal):
+
+        # Odd eigenvectors: negative direction, even eigenvectors: positive direction
+        df_odd = replicas_df.loc[:, 2::2]
+        df_even = replicas_df.loc[:, 3::2]
+        new_column_names = range(1, len(df_even.columns) + 1)
+
+        df_even.columns = new_column_names
+        df_odd.columns = new_column_names
+
+        central_member, hess_diff_cov = replicas_df.loc[:, [1]], df_even - df_odd
+
+        # Eq. 4.3 of arXiv:2203.05506
+        mc_replica = central_member.dot([1]) + 0.5 * hess_diff_cov.dot(rnd_std_norm_vec)
+
+        headers = f"PdfType: replica\nFormat: lhagrid1\nFromHessReplica: {i}\n"
+        write_replica(i + 1, mc_pdf_path, headers.encode("UTF-8"), mc_replica)
+
+    # Write central replica from hessian set to mc set
+    headers = f"PdfType: replica\nFormat: lhagrid1\nFromHessReplica: {i}\n"
+    log.info(f"Writing central replica to {mc_pdf_path}")
+    write_replica(0, mc_pdf_path, headers.encode("UTF-8"), central_member)
+
+
+@check_pdf_is_hessian
+def write_hessian_to_mc_watt_thorne(pdf, mc_pdf_name, num_members, watt_thorne_rnd_seed=1):
+    """
+    Writes the Monte Carlo representation of a PDF set that is in Hessian form
+    using the Watt-Thorne (MSHT20) prescription described in Eq. 4.3 of arXiv:2203.05506.
+
+    Parameters
+    ----------
+    pdf: validphys.core.PDF
+        The Hessian PDF set that is to be converted to Monte Carlo.
+
+    mc_pdf_name: str
+        The name of the new Monte Carlo PDF set.
+
+    """
+    hessian_set = pdf
+
+    lhapdf_path = pathlib.Path(lhapdf.paths()[-1])
+
+    # path to hessian lhapdf set
+    hessian_pdf_path = lhapdf_path / str(hessian_set)
+
+    # path to new pdf set
+    mc_pdf_path = lhapdf_path / mc_pdf_name
+
+    # create new pdf set folder in lhapdf path if it does not exist
+    if not mc_pdf_path.exists():
+        os.makedirs(mc_pdf_path)
+
+    # write LHAPDF info file for new pdf set
+    write_new_lhapdf_info_file_from_previous_pdf(
+        path_old_pdfset=hessian_pdf_path,
+        name_old_pdfset=str(hessian_set),
+        path_new_pdfset=mc_pdf_path,
+        name_new_pdfset=mc_pdf_name,
+        num_members=num_members,
+        description_set=f"MC representation of {str(hessian_set)}",
+        errortype="replicas",
+    )
+
+    # load replicas from basis set at all scales
+    _, grids = load_all_replicas(hessian_set)
+    replicas_df = rep_matrix(grids)
+
+    # Eg for MSHT20, mem=0 => central value; mem=1-64 => 32 eigenvector sets (+/- directions)
+    n_eig = int((replicas_df.shape[1] - 1) / 2)
+
+    np.random.seed(watt_thorne_rnd_seed)
+    Rjk_std_normal = np.random.standard_normal(size=(num_members, n_eig))
+
+    # write replicas to new pdf set
+    write_mc_watt_thorne_replicas(Rjk_std_normal, replicas_df, mc_pdf_path)
diff --git a/validphys2/src/validphys/tests/test_hessian2mc.py b/validphys2/src/validphys/tests/test_hessian2mc.py
new file mode 100644
index 0000000000..68f55fa695
--- /dev/null
+++ b/validphys2/src/validphys/tests/test_hessian2mc.py
@@ -0,0 +1,82 @@
+import numpy as np
+import pandas as pd
+from unittest import mock
+from validphys.hessian2mc import write_mc_watt_thorne_replicas, write_hessian_to_mc_watt_thorne
+import pathlib
+
+
+@mock.patch("validphys.hessian2mc.write_replica")
+@mock.patch("validphys.hessian2mc.log.info")
+def test_write_mc_watt_thorne_replicas(mock_log_info, mock_write_replica):
+    # Set up mock inputs
+    n_eig = 5
+    n_all_eig = 1 + 2 * n_eig  # includes central member, so 1 + 2*n_eig
+    num_members = 50  # new MC members
+    num_nx_nq = 100  # xgrid and Q points
+
+    replicas_df = pd.DataFrame(
+        np.random.standard_normal(size=(num_nx_nq, n_all_eig)), columns=range(1, n_all_eig + 1)
+    )
+
+    Rjk_std_normal = np.random.standard_normal(size=(num_members, n_eig))
+
+    # import IPython; IPython.embed()
+    mc_pdf_path = mock.Mock()  # Mock the path
+
+    # Call the function being tested
+    write_mc_watt_thorne_replicas(Rjk_std_normal, replicas_df, mc_pdf_path)
+
+    # Verify that write_replica was called the correct number of times
+    assert (
+        mock_write_replica.call_count == num_members + 1
+    )  # for Rjk members + 1 for the central replica
+
+    # Check if the log message was correct for the central replica
+    mock_log_info.assert_any_call(f"Writing central replica to {mc_pdf_path}")
+
+
+@mock.patch("validphys.hessian2mc.write_mc_watt_thorne_replicas")
+@mock.patch("validphys.hessian2mc.load_all_replicas")
+@mock.patch("validphys.hessian2mc.rep_matrix")
+@mock.patch("validphys.hessian2mc.write_new_lhapdf_info_file_from_previous_pdf")
+@mock.patch("validphys.hessian2mc.os.makedirs")
+@mock.patch("validphys.hessian2mc.lhapdf.paths")
+def test_write_hessian_to_mc_watt_thorne(
+    mock_lhapdf_paths,
+    mock_makedirs,
+    mock_write_info_file,
+    mock_rep_matrix,
+    mock_load_all_replicas,
+    mock_write_mc_replicas,
+):
+    # Set up mock inputs
+    msht_like_hessian_pdf = "MSHT20"
+    mc_pdf_name = "MSHT20_MC"
+    num_members = 100
+
+    mock_load_all_replicas.return_value = (None, None)
+
+    mock_lhapdf_paths.return_value = [pathlib.Path("/path/to/lhapdf")]
+
+    mock_rep_matrix.return_value = np.random.randn(5, 7)  # Mocked replica matrix
+
+    # Call the function being tested
+    write_hessian_to_mc_watt_thorne(msht_like_hessian_pdf, mc_pdf_name, num_members)
+
+    # Verify that the necessary directories were created
+    mc_pdf_path = pathlib.Path("/path/to/lhapdf") / mc_pdf_name
+    mock_makedirs.assert_called_once_with(mc_pdf_path)
+
+    # Verify that the LHAPDF info file was written
+    mock_write_info_file.assert_called_once_with(
+        path_old_pdfset=pathlib.Path("/path/to/lhapdf") / msht_like_hessian_pdf,
+        name_old_pdfset=msht_like_hessian_pdf,
+        path_new_pdfset=mc_pdf_path,
+        name_new_pdfset=mc_pdf_name,
+        num_members=num_members,
+        description_set=f"MC representation of {msht_like_hessian_pdf}",
+        errortype="replicas",
+    )
+
+    # Verify that the replicas were written
+    mock_write_mc_replicas.assert_called_once()