diff --git a/python/tests/test_haplotype_matching.py b/python/tests/test_haplotype_matching.py
index dcc1d684fb..7b0786663e 100644
--- a/python/tests/test_haplotype_matching.py
+++ b/python/tests/test_haplotype_matching.py
@@ -1,6 +1,6 @@
 # MIT License
 #
-# Copyright (c) 2019-2023 Tskit Developers
+# Copyright (c) 2019-2024 Tskit Developers
 #
 # Permission is hereby granted, free of charge, to any person obtaining a copy
 # of this software and associated documentation files (the "Software"), to deal
@@ -484,6 +484,30 @@ def run(self, h, normalisation_factor):
         return self.output
 
 
+class ForwardBackwardAlgorithm(BackwardAlgorithm):
+    """
+    The Li and Stephens forward-backward algorithm.
+
+    Alternatively, it is more efficient to compute this
+    on the fly during a run of the backward algorithm.
+    """
+
+    def compute_next_probability(self, site_id, p_next, is_match, node):
+        p_e = self.compute_emission_proba(site_id, is_match)
+        bwd_prob = p_next * p_e
+        # Get forward prob. from compressed matrix.
+        fwd_prob = 0
+        return bwd_prob * fwd_prob
+
+    def run(self, h, normalisation_factor):
+        self.initialise(value=1)
+        while self.tree.prev():
+            self.update_tree(direction=tskit.REVERSE)
+            for site in reversed(list(self.tree.sites())):
+                self.process_site(site, h[site.id], normalisation_factor[site.id])
+        return self.output
+
+
 class ViterbiAlgorithm(LsHmmAlgorithm):
     """
     Runs the Li and Stephens Viterbi algorithm.
@@ -729,6 +753,40 @@ def ls_backward_tree(h, ts, rho, mu, normalisation_factor, precision=30, alleles
     return ba.run(h, normalisation_factor)
 
 
+def ls_fb_tree(
+    h, ts, rho, mu, precision=30, alleles=None, scale_mutation_based_on_n_alleles=False
+):
+    alleles, n_alleles = get_site_alleles(ts, h, alleles)
+    fa = ForwardAlgorithm(
+        ts,
+        rho,
+        mu,
+        alleles,
+        n_alleles,
+        precision=precision,
+        scale_mutation=scale_mutation_based_on_n_alleles,
+    )
+    forward_cm = fa.run(h)
+    ba = BackwardAlgorithm(
+        ts,
+        rho,
+        mu,
+        alleles,
+        n_alleles,
+        precision=precision,
+    )
+    _ = ba.run(h, forward_cm.normalisation_factor)
+    fba = ForwardBackwardAlgorithm(
+        ts,
+        rho,
+        mu,
+        alleles,
+        n_alleles,
+        precision=precision,
+    )
+    return fba.run(h)
+
+
 def ls_viterbi_tree(
     h, ts, rho, mu, precision=30, alleles=None, scale_mutation_based_on_n_alleles=False
 ):
@@ -1184,6 +1242,10 @@ def check_backward_matrix(ts, h, forward_cm, recombination=None, mutation=None):
     nt.assert_allclose(B, B_lib)
 
 
+def check_fb_matrix():
+    raise NotImplementedError()
+
+
 def add_unique_sample_mutations(ts, start=0):
     """
     Adds a mutation for each of the samples at equally spaced locations