diff --git a/executor/src/witgen/global_constraints.rs b/executor/src/witgen/global_constraints.rs
index 745810cd7..c32df2477 100644
--- a/executor/src/witgen/global_constraints.rs
+++ b/executor/src/witgen/global_constraints.rs
@@ -5,7 +5,8 @@ use num_traits::Zero;
 
 use powdr_ast::analyzed::{
     AlgebraicBinaryOperation, AlgebraicBinaryOperator, AlgebraicExpression as Expression,
-    AlgebraicReference, LookupIdentity, PermutationIdentity, PolyID, PolynomialType,
+    AlgebraicReference, LookupIdentity, PermutationIdentity, PhantomLookupIdentity,
+    PhantomPermutationIdentity, PolyID, PolynomialType,
 };
 
 use powdr_number::FieldElement;
@@ -236,7 +237,9 @@ fn propagate_constraints<T: FieldElement>(
             }
         }
         Identity::Lookup(LookupIdentity { left, right, .. })
-        | Identity::Permutation(PermutationIdentity { left, right, .. }) => {
+        | Identity::PhantomLookup(PhantomLookupIdentity { left, right, .. })
+        | Identity::Permutation(PermutationIdentity { left, right, .. })
+        | Identity::PhantomPermutation(PhantomPermutationIdentity { left, right, .. }) => {
             if left.selector.is_some() || right.selector.is_some() {
                 return (known_constraints, false);
             }
@@ -253,7 +256,7 @@ fn propagate_constraints<T: FieldElement>(
                 }
             }
             if right.expressions.len() == 1 {
-                // We can only remove the lookup if the RHS is a fixed polynomial that
+                // We can only remove the lookup or permutation if the RHS is a fixed polynomial that
                 // provides all values in the span.
                 if let Some(name) = try_to_simple_poly(&right.expressions[0]) {
                     if try_to_simple_poly(&left.expressions[0]).is_some()
@@ -264,9 +267,6 @@ fn propagate_constraints<T: FieldElement>(
                 }
             }
         }
-        Identity::PhantomPermutation(..) | Identity::PhantomLookup(..) => {
-            // phantom identities do not provide any constraints
-        }
         Identity::Connect(..) => {
             // we do not handle connect identities yet, so we do nothing
         }
diff --git a/executor/src/witgen/machines/machine_extractor.rs b/executor/src/witgen/machines/machine_extractor.rs
index dac58d632..7f5d42984 100644
--- a/executor/src/witgen/machines/machine_extractor.rs
+++ b/executor/src/witgen/machines/machine_extractor.rs
@@ -349,7 +349,9 @@ fn refs_in_selected_expressions<T>(sel_expr: &SelectedExpressions<T>) -> HashSet
 fn refs_in_identity_left<T>(identity: &Identity<T>) -> HashSet<PolyID> {
     match identity {
         Identity::Lookup(LookupIdentity { left, .. })
-        | Identity::Permutation(PermutationIdentity { left, .. }) => {
+        | Identity::PhantomLookup(PhantomLookupIdentity { left, .. })
+        | Identity::Permutation(PermutationIdentity { left, .. })
+        | Identity::PhantomPermutation(PhantomPermutationIdentity { left, .. }) => {
             refs_in_selected_expressions(left)
         }
         Identity::Polynomial(i) => refs_in_expression(&i.expression).collect(),