diff --git a/NEWS b/NEWS
index 23c15f3d1..ff2925ce6 100644
--- a/NEWS
+++ b/NEWS
@@ -110,6 +110,11 @@ New in spot 2.5.3.dev  (not yet released)
     The code was buggy, hard to maintain, and these functions do not
     seem to be used.
 
+  - spot::remove_alternation() was slightly improved on very-weak
+    alternating automata: the labeling of the outgoing transitions in
+    the resulting TGBA makes it more likely that simulation-based
+    reductions will reduce it.
+
   Python:
 
   - New spot.jupyter package.  This currently contains a function for
diff --git a/spot/twaalgos/alternation.cc b/spot/twaalgos/alternation.cc
index 40f0a2a42..290e80f8a 100644
--- a/spot/twaalgos/alternation.cc
+++ b/spot/twaalgos/alternation.cc
@@ -184,6 +184,7 @@ namespace spot
       std::map<int, unsigned> var_to_state_;
       std::vector<int> scc_to_var_;
       std::map<int, acc_cond::mark_t> var_to_mark_;
+      std::vector<unsigned> mark_to_state_;
       bdd all_vars_;
       bdd all_marks_;
       bdd all_states_;
@@ -223,6 +224,7 @@ namespace spot
         unsigned nc = si_.scc_count();
         scc_to_var_.reserve(nc);
         unsigned mark_pos = has_reject_more_;
+        mark_to_state_.resize(mark_pos);
         bdd all_marks = bddtrue;
         for (unsigned s = 0; s < nc; ++s)
           {
@@ -231,6 +233,7 @@ namespace spot
               {
                 int v = d->register_anonymous_variables(1, this);
                 scc_to_var_.emplace_back(v);
+                mark_to_state_.push_back(si_.one_state_of(c));
                 var_to_mark_.emplace(v, acc_cond::mark_t({mark_pos++}));
                 bdd bv = bdd_ithvar(v);
                 all_marks &= bv;
@@ -270,7 +273,7 @@ namespace spot
                   marked && (scc_s == scc_d) && (c == scc_class::reject_more);
                 dest &= bdd_ithvar(state_to_var_[d] + mark);
                 if (c == scc_class::reject_1 && scc_s == scc_d)
-                  dest &= bdd_ithvar(scc_to_var_[scc_s]);
+                  dest &= bdd_ithvar(scc_to_var_[scc_d]);
               }
             res |= e.cond & dest;
           }
@@ -464,6 +467,29 @@ namespace spot
                     bdd dest = bdd_existcomp(cube, all_vars_);
                     v.clear();
                     acc_cond::mark_t m = bdd_to_state(dest, v);
+
+                    // if there is no promise "f" is between a state
+                    // that does not have f, and a state that have
+                    // "f", we can add one.  Doing so will help later
+                    // simplifications performed by postprocessor.  An
+                    // example where this is needed is the VWAA
+                    // generated by ltl[23]ba for GFa.  Without the
+                    // next loop, the final TGBA has 2 states instead
+                    // of 1.
+                    for (unsigned m1: (all_marks - m).sets())
+                      {
+                        if (has_reject_more_ && m1 == 0)
+                          continue;
+                        auto& sv = s_to_ss[s];
+                        unsigned ms = mark_to_state_[m1];
+                        if (std::find(v.begin(), v.end(), ms) != v.end())
+                          {
+                            unsigned ms = mark_to_state_[m1];
+                            if (std::find(sv.begin(), sv.end(), ms) == sv.end())
+                              m.set(m1);
+                          }
+                      }
+
                     unsigned d = new_state(v, has_mark);
                     if (has_mark)
                       m.set(0);
diff --git a/tests/core/alternating.test b/tests/core/alternating.test
index feb95aa59..6a1798f0f 100755
--- a/tests/core/alternating.test
+++ b/tests/core/alternating.test
@@ -195,7 +195,7 @@ Acceptance: 1 Inf(0)
 properties: trans-labels explicit-labels trans-acc complete
 --BODY--
 State: 0
-[t] 1 {0}
+[t] 1
 State: 1
 [t] 1
 [0] 1 {0}
diff --git a/tests/python/alternation.ipynb b/tests/python/alternation.ipynb
index c96896f59..10be77db2 100644
--- a/tests/python/alternation.ipynb
+++ b/tests/python/alternation.ipynb
@@ -1373,7 +1373,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [
     {
@@ -1961,7 +1961,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
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@@ -2825,8 +2825,8 @@
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@@ -2840,10 +2840,9 @@
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@@ -2854,11 +2853,9 @@
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@@ -2874,15 +2871,13 @@
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