diff --git a/NEWS b/NEWS
index 0a403b473..c9efc427d 100644
--- a/NEWS
+++ b/NEWS
@@ -215,6 +215,9 @@ New in spot 2.3.5.dev (not yet released)
     generalized co-Büchi), it would sometimes output an automaton with
     transition-based acceptance but marked as state-based.
 
+  - The complete() function could complete an empty co-Büchi automaton into an
+    automaton accepting everything.
+
   Backward-incompatible changes:
 
   - spot::acc_cond::mark_t::operator bool() has been marked as
diff --git a/spot/twaalgos/complete.cc b/spot/twaalgos/complete.cc
index 2373eeaf8..71426c795 100644
--- a/spot/twaalgos/complete.cc
+++ b/spot/twaalgos/complete.cc
@@ -69,7 +69,7 @@ namespace spot
                     loopmark = t.acc;
                     first = false;
                   }
-                // If this this not the first self loop and it as a
+                // If this this not the first self loop and it has a
                 // different acceptance mark, do not consider this
                 // state as a sink candidate: combining loops with
                 // different marks might be used to build an accepting
@@ -107,7 +107,7 @@ namespace spot
     for (unsigned i = 0; i < n; ++i)
       {
         bdd missingcond = bddtrue;
-        acc_cond::mark_t acc = 0U;
+        acc_cond::mark_t acc = um.second;
         unsigned edge_to_sink = 0;
         for (auto& t: aut->out(i))
           {
@@ -121,6 +121,8 @@ namespace spot
             // In case the automaton uses edge-based acceptance,
             // it does not matter what acceptance set we put the new
             // edge into.
+            // EXCEPT if there is an incomplete sinkable state: completing it
+            // creates self-loops that must be rejecting.
             //
             // So in both cases, we put the edge in the same
             // acceptance sets as the last outgoing edge of the
@@ -147,10 +149,11 @@ namespace spot
               }
             else                // otherwise, create the new edge.
               {
-                // in case the automaton use state-based acceptance, propagate
+                // in case the automaton uses state-based acceptance, propagate
                 // the acceptance of the first edge to the one we add.
-                if (!aut->prop_state_acc().is_true())
+                if (!aut->prop_state_acc().is_true() && i != sink)
                   acc = 0U;
+
                 aut->new_edge(i, sink, missingcond, acc);
               }
           }
diff --git a/tests/core/complete.test b/tests/core/complete.test
index 6b038dd09..80d284968 100755
--- a/tests/core/complete.test
+++ b/tests/core/complete.test
@@ -25,6 +25,14 @@ set -e
 
 cat >automaton <<EOF
 HOA: v1
+States: 1
+Start: 0
+AP: 1 "a"
+Acceptance: 1 Fin(0)
+--BODY--
+State: 0
+--END--
+HOA: v1
 States: 2
 Start: 0
 AP: 1 "a"
@@ -107,6 +115,18 @@ EOF
 
 cat >expected <<EOF
 HOA: v1
+States: 1
+Start: 0
+AP: 1 "a"
+acc-name: co-Buchi
+Acceptance: 1 Fin(0)
+properties: trans-labels explicit-labels state-acc colored complete
+properties: deterministic
+--BODY--
+State: 0 {0}
+[t] 0
+--END--
+HOA: v1
 States: 3
 Start: 0
 AP: 1 "a"
@@ -210,12 +230,12 @@ Start: 0&1
 AP: 0
 acc-name: co-Buchi
 Acceptance: 1 Fin(0)
-properties: trans-labels explicit-labels state-acc complete
+properties: trans-labels explicit-labels state-acc colored complete
 properties: deterministic univ-branch
 --BODY--
-State: 0
+State: 0 {0}
 [t] 0
-State: 1
+State: 1 {0}
 [t] 0
 --END--
 EOF
diff --git a/tests/core/prodor.test b/tests/core/prodor.test
index 35640d6cb..3684aa5ec 100755
--- a/tests/core/prodor.test
+++ b/tests/core/prodor.test
@@ -24,6 +24,32 @@ set -e
 ltl2tgba=ltl2tgba
 autfilt=autfilt
 
+# Two empty automata
+cat >a1.hoa <<EOF
+HOA: v1
+States: 1
+Start: 0
+AP: 1 "a"
+Acceptance: 1 Fin(0)
+--BODY--
+State: 0
+--END--
+EOF
+cat >a2.hoa <<EOF
+HOA: v1
+States: 1
+Start: 0
+AP: 1 "a"
+Acceptance: 0 f
+--BODY--
+State: 0
+[t] 0
+--END--
+EOF
+
+# the OR product of two empty automata should be empty
+$autfilt --product-or a1.hoa a2.hoa -H | $autfilt --is-empty
+
 $ltl2tgba -DH 'GFa' > gfa.hoa
 $ltl2tgba -DH 'FGb' > fgb.hoa
 $autfilt --product-or gfa.hoa fgb.hoa -H > por.hoa