diff --git a/src/subtype.c b/src/subtype.c
index 5edcd100ee8e03..f5c13b77ea0cf5 100644
--- a/src/subtype.c
+++ b/src/subtype.c
@@ -2464,8 +2464,10 @@ static jl_value_t *intersect_aside(jl_value_t *x, jl_value_t *y, jl_stenv_t *e,
         return y;
     if (y == (jl_value_t*)jl_any_type && !jl_is_typevar(x))
         return x;
-    // band-aid for #46736
-    if (obviously_egal(x, y))
+    // band-aid for #46736 #56040
+    if (obviously_in_union(x, y))
+        return y;
+    if (obviously_in_union(y, x))
         return x;
 
     jl_varbinding_t *vars = NULL;
@@ -2495,6 +2497,9 @@ static jl_value_t *intersect_aside(jl_value_t *x, jl_value_t *y, jl_stenv_t *e,
 
 static jl_value_t *intersect_union(jl_value_t *x, jl_uniontype_t *u, jl_stenv_t *e, int8_t R, int param)
 {
+    // band-aid for #56040
+    if (!jl_is_uniontype(x) && obviously_in_union((jl_value_t *)u, x))
+        return x;
     int no_free = !jl_has_free_typevars(x) && !jl_has_free_typevars((jl_value_t*)u);
     if (param == 2 || no_free) {
         jl_value_t *a=NULL, *b=NULL;
diff --git a/test/subtype.jl b/test/subtype.jl
index 7be869107b432a..dfa1487eaa55da 100644
--- a/test/subtype.jl
+++ b/test/subtype.jl
@@ -2721,3 +2721,12 @@ let T1 = NTuple{12, Union{Val{1}, Val{2}, Val{3}, Val{4}, Val{5}, Val{6}}}
     @test !(T1 <: T2)
     @test Tuple{Union{Val{1},Val{2}}} <: Tuple{S} where {T, S<:Val{T}}
 end
+
+#issue 56040
+let S = Dict{V,V} where {V},
+    T = Dict{Ref{Union{Set{A2}, Set{A3}, A3}}, Ref{Union{Set{A3}, Set{A2}, Set{A1}, Set{A4}, A4}}} where {A1, A2<:Set{A1}, A3<:Union{Set{A1}, Set{A2}}, A4<:Union{Set{A2}, Set{A1}, Set{A3}}},
+    A = Dict{Ref{Set{Union{}}}, Ref{Set{Union{}}}}
+    @testintersect(S, T, !Union{})
+    @test A <: typeintersect(S, T)
+    @test A <: typeintersect(T, S)
+end