diff --git a/compiler/rustc_infer/src/infer/outlives/test_type_match.rs b/compiler/rustc_infer/src/infer/outlives/test_type_match.rs
index de42736ce2b06..e6ca880bd18c6 100644
--- a/compiler/rustc_infer/src/infer/outlives/test_type_match.rs
+++ b/compiler/rustc_infer/src/infer/outlives/test_type_match.rs
@@ -44,9 +44,6 @@ pub fn extract_verify_if_eq<'tcx>(
     assert!(!verify_if_eq_b.has_escaping_bound_vars());
     let mut m = MatchAgainstHigherRankedOutlives::new(tcx);
     let verify_if_eq = verify_if_eq_b.skip_binder();
-    debug_assert!(
-        !tcx.next_trait_solver_globally() || !(verify_if_eq.ty, test_ty).has_non_rigid_aliases()
-    );
     m.relate(verify_if_eq.ty, test_ty).ok()?;
 
     if let ty::RegionKind::ReBound(index_kind, br) = verify_if_eq.bound.kind() {
@@ -198,6 +195,13 @@ impl<'tcx> TypeRelation<TyCtxt<'tcx>> for MatchAgainstHigherRankedOutlives<'tcx>
             self.no_match()
         } else if pattern == value {
             Ok(pattern)
+        } else if let (ty::Alias(_, pattern_alias), ty::Alias(_, value_alias)) =
+            (*pattern.kind(), *value.kind())
+        {
+            // Rigidness is normalization state. This matcher only needs to know
+            // whether both sides are the same alias shape.
+            self.relate(pattern_alias, value_alias)?;
+            Ok(pattern)
         } else {
             relate::structurally_relate_tys(self, pattern, value)
         }
@@ -211,6 +215,13 @@ impl<'tcx> TypeRelation<TyCtxt<'tcx>> for MatchAgainstHigherRankedOutlives<'tcx>
     ) -> RelateResult<'tcx, ty::Const<'tcx>> {
         if pattern == value {
             Ok(pattern)
+        } else if let (
+            ty::ConstKind::Alias(_, pattern_alias),
+            ty::ConstKind::Alias(_, value_alias),
+        ) = (pattern.kind(), value.kind())
+        {
+            self.relate(pattern_alias, value_alias)?;
+            Ok(pattern)
         } else {
             relate::structurally_relate_consts(self, pattern, value)
         }
diff --git a/compiler/rustc_trait_selection/src/traits/outlives_for_liveness.rs b/compiler/rustc_trait_selection/src/traits/outlives_for_liveness.rs
index 5cba32d742f62..ad80864db924a 100644
--- a/compiler/rustc_trait_selection/src/traits/outlives_for_liveness.rs
+++ b/compiler/rustc_trait_selection/src/traits/outlives_for_liveness.rs
@@ -56,10 +56,6 @@ pub(crate) fn live_args_for_alias_from_outlives_bounds<'tcx>(
                 test_type_match::extract_verify_if_eq(
                     tcx,
                     &outlives.map_bound(|ty::OutlivesClause(ty, bound)| VerifyIfEq { ty, bound }),
-                    // FIXME(#155345): Region handling should generally only
-                    // deal with rigid aliases, making sure we do so correctly
-                    // everywhere is effort, so we're just using `No` everywhere
-                    // for now. This should change soon.
                     alias_ty,
                 )
             }
diff --git a/tests/ui/traits/next-solver/rigid-alias-liveness-issue-160206.rs b/tests/ui/traits/next-solver/rigid-alias-liveness-issue-160206.rs
new file mode 100644
index 0000000000000..4899645e10d58
--- /dev/null
+++ b/tests/ui/traits/next-solver/rigid-alias-liveness-issue-160206.rs
@@ -0,0 +1,27 @@
+//@ compile-flags: -Znext-solver=globally
+//@ check-pass
+
+trait Foo<'x> {
+    type Out;
+    fn foo(self) -> Self::Out;
+}
+
+struct Bar;
+
+impl<'x> Foo<'x> for Bar {
+    type Out = ();
+
+    fn foo(self) -> Self::Out {
+        todo!()
+    }
+}
+
+fn make_static_foo<'x>(_: &'x ()) -> impl Foo<'x, Out: 'static> {
+    Bar
+}
+
+fn test() {
+    make_static_foo(&());
+}
+
+fn main() {}