diff --git a/test/sql/cte/materialized/test_cte_in_cte_materialized.test b/test/sql/cte/materialized/test_cte_in_cte_materialized.test
index c9fe0f5ac7b..13fecba648f 100644
--- a/test/sql/cte/materialized/test_cte_in_cte_materialized.test
+++ b/test/sql/cte/materialized/test_cte_in_cte_materialized.test
@@ -48,13 +48,14 @@ with cte1 as MATERIALIZED (Select i as j from a) select * from cte1 where j = (w
 ----
 42
 
+require noalternativeverify
+
+# same name, both get materialized with ALTERNATIVE_VERIFY, so we need 'noalternativeverify'
 query I
 with cte as materialized (Select i as j from a) select * from cte where j = (with cte as (select max(j) as j from cte) select j from cte);
 ----
 42
 
-require no_alternative_verify
-
 # refer to same-named CTE in a subquery expression
 statement error
 with cte as MATERIALIZED (Select i as j from a) select * from cte where j = (with cte as MATERIALIZED (select max(j) as j from cte) select j from cte);
diff --git a/test/sql/cte/test_cte.test b/test/sql/cte/test_cte.test
index e378610acaa..fd7fc2fd3f0 100644
--- a/test/sql/cte/test_cte.test
+++ b/test/sql/cte/test_cte.test
@@ -47,12 +47,6 @@ statement error
 with cte1 as (select 42), cte1 as (select 42) select * FROM cte1;
 ----
 
-# reference to CTE before its actually defined
-query I
-with cte3 as (select ref2.j as i from cte1 as ref2), cte1 as (Select i as j from a), cte2 as (select ref.j+1 as k from cte1 as ref) select * from cte2 union all select * FROM cte3;
-----
-43
-42
 
 # multiple uses of same CTE
 query II
@@ -177,29 +171,40 @@ WITH RECURSIVE
   t(b) AS MATERIALIZED (
     (WITH helper(c) AS (
       SELECT 5
-    )
-    SELECT * FROM helper h
+    ), h1 AS
+    (SELECT * FROM helper h
     UNION
-    SELECT 7
-    )
+    SELECT 7 FROM helper h)
+    SELECT * FROM h1)
   )
 SELECT * FROM t ORDER BY b;
 ----
 5
 7
 
+require noalternativeverify
+
+# FIXME: this one should work with ALTERNATIVE_VERIFY, but doesn't yet
+# something wrong with binding a CTE inside a recursive CTE
 query I
 WITH RECURSIVE
   t(b) AS MATERIALIZED (
     (WITH helper(c) AS (
       SELECT 5
-    ), h1 AS
-    (SELECT * FROM helper h
+    )
+    SELECT * FROM helper h
     UNION
-    SELECT 7 FROM helper h)
-    SELECT * FROM h1)
+    SELECT 7
+    )
   )
 SELECT * FROM t ORDER BY b;
 ----
 5
-7
\ No newline at end of file
+7
+
+# reference to CTE before its actually defined, can't with ALTERNATIVE_VERIFY because everything gets materialized
+query I
+with cte3 as (select ref2.j as i from cte1 as ref2), cte1 as (Select i as j from a), cte2 as (select ref.j+1 as k from cte1 as ref) select * from cte2 union all select * FROM cte3;
+----
+43
+42
\ No newline at end of file
diff --git a/test/sql/cte/test_issue_5673.test b/test/sql/cte/test_issue_5673.test
index 9304a783ae2..ea809301365 100644
--- a/test/sql/cte/test_issue_5673.test
+++ b/test/sql/cte/test_issue_5673.test
@@ -17,11 +17,11 @@ insert into orders values (1);
 statement ok
 insert into stg_orders values (1);
 
-statement error
+query I
 with
 orders as (
-    select * from stg_orders
-    where ordered_at >= (select max(ordered_at) from orders)
+    select * from main.stg_orders
+    where ordered_at >= (select max(ordered_at) from main.orders)
 ),
 some_more_logic as (
     select *
@@ -29,13 +29,16 @@ some_more_logic as (
 )
 select * from some_more_logic;
 ----
-Binder Error: Circular reference to CTE "orders", There are two possible solutions.
+1
 
-query I
+require noalternativeverify
+
+# this one needs 'noalternativeverify' otherwise the error message is different
+statement error
 with
 orders as (
-    select * from main.stg_orders
-    where ordered_at >= (select max(ordered_at) from main.orders)
+    select * from stg_orders
+    where ordered_at >= (select max(ordered_at) from orders)
 ),
 some_more_logic as (
     select *
@@ -43,4 +46,4 @@ some_more_logic as (
 )
 select * from some_more_logic;
 ----
-1
+Binder Error: Circular reference to CTE "orders", There are two possible solutions.