diff --git a/src/bin/autfilt.cc b/src/bin/autfilt.cc
index db9cda1ae..c814e384a 100644
--- a/src/bin/autfilt.cc
+++ b/src/bin/autfilt.cc
@@ -415,12 +415,16 @@ namespace
       // Do this first, because it is cheap and will help most
       // algorithms.
       if (opt_merge)
-	aut->merge_transitions();
+        {
+          aut->merge_transitions();
+          if (opt_are_isomorphic)
+            opt_are_isomorphic->merge_transitions();
+        }
 
       if (opt_product)
 	aut = spot::product(std::move(aut), opt_product);
 
-      if (opt_isomorph && !are_isomorphic(aut, opt_isomorph))
+      if (opt_isomorph && are_isomorphic(aut, opt_isomorph).empty())
         return 0;
 
       aut = post.run(aut, nullptr);
diff --git a/src/tgbaalgos/are_isomorphic.cc b/src/tgbaalgos/are_isomorphic.cc
index 0f3a7c026..86e3a2692 100644
--- a/src/tgbaalgos/are_isomorphic.cc
+++ b/src/tgbaalgos/are_isomorphic.cc
@@ -187,12 +187,12 @@ namespace
 
 namespace spot
 {
-  bool
+  std::vector<unsigned>
   are_isomorphic(const const_tgba_digraph_ptr a1,
                  const const_tgba_digraph_ptr a2)
   {
     if (are_trivially_different(a1, a2))
-      return false;
+      return std::vector<unsigned>();
     unsigned n = a1->num_states();
     assert(n == a2->num_states());
     class2states_t a1_classes = map_class_states(a1);
@@ -202,14 +202,14 @@ namespace spot
     // Get the first possible mapping between a1 and a2, or return false if
     // the number of class or the size of the classes do not match.
     if (!(mapping_from_classes(mapping, a1_classes, a2_classes)))
-      return false;
+      return std::vector<unsigned>();
 
     while (!is_isomorphism(mapping, a1, a2))
       {
         if (!next_class_permutation(a2_classes))
-          return false;
+          return std::vector<unsigned>();
         mapping_from_classes(mapping, a1_classes, a2_classes);
       }
-    return true;
+    return mapping;
   }
 }
diff --git a/src/tgbaalgos/are_isomorphic.hh b/src/tgbaalgos/are_isomorphic.hh
index e4b3b38ff..9e8273754 100644
--- a/src/tgbaalgos/are_isomorphic.hh
+++ b/src/tgbaalgos/are_isomorphic.hh
@@ -21,20 +21,16 @@
 # define SPOT_TGBAALGOS_ARE_ISOMORPHIC_HH
 
 #include "tgba/tgbagraph.hh"
+#include <vector>
 
 namespace spot
 {
   /// \ingroup tgba_misc
-  /// \brief Check whether two automata are isomorphic.
-  /// Two automata a1 and a2 are said to be isomorphic if there is a bijection
-  /// f between the states of a1 and a2 such that for every pair of state (s1,
-  /// s2) of a1:
-  /// - there is a transition from s1 to s2 iff there is a transition from f(s1)
-  /// to f(s2)
-  /// - if there is such a transition, then the acceptance set and acceptance
-  /// condition are the same on the transition from s1 to s2 and from f(s1) to
-  /// f(s2)
-  SPOT_API bool are_isomorphic(const const_tgba_digraph_ptr a1,
+  /// \brief Return an isomorphism between a1 and a2 if such an isomorphism
+  /// exists. Otherwise, return an empty vector.
+  /// The return value is a vector indexed by states of a1, and containing
+  /// states of a2.
+  SPOT_API std::vector<unsigned> are_isomorphic(const const_tgba_digraph_ptr a1,
                                const const_tgba_digraph_ptr a2);
 }