diff --git a/packages/downgrader/README.md b/packages/downgrader/README.md
index cab1d34..6ccf377 100644
--- a/packages/downgrader/README.md
+++ b/packages/downgrader/README.md
@@ -242,6 +242,8 @@ Accepts the default OAS dialect, which adds `discriminator`, `xml`, `externalDoc
 
 - A `not` or `oneOf` inside a cycle of schemas, over a `$ref` back into that cycle to a schema that lost a restriction, can still reject values the original accepts.
 - `$ref`s to an `$anchor`, or written relative to an `$id`, are left as written and dangle. Every `$ref` is read as a JSON Pointer from the document root, and one into `$defs` is recognized by its `/$defs/` segment.
+- An `anyOf` of `{ type: "null" }` and a branch that a `$ref` points into stays as written, so the `$ref` still resolves, and code generators read it as a union.
+- `$ref`s into a keyword 3.0 removes, such as `prefixItems/0` or `propertyNames`, or into a `oneOf` that becomes `anyOf`, are left as written and dangle.
 - Recursion through `$defs` is cut to `{}` after one level. Where several `$ref`s enter the same cycle, the first copy made is reused, so a later one can be cut sooner.
 - Older drafts are not supported. `definitions` is read as `$defs`, but other keywords that only draft-07 or 2019-09 define, such as `dependencies` and `additionalItems`, become extensions unconverted, so `$ref`s into them dangle, and array-form `items` passes through. Convert such schemas to 2020-12 first.
 - `patternProperties` that matches every name, as TypeBox emits for a record, is removed with its value schema rather than turned into `additionalProperties`.
diff --git a/packages/downgrader/src/v3.1-to-v3.0.ts b/packages/downgrader/src/v3.1-to-v3.0.ts
index 2a0fe60..742c404 100644
--- a/packages/downgrader/src/v3.1-to-v3.0.ts
+++ b/packages/downgrader/src/v3.1-to-v3.0.ts
@@ -127,7 +127,11 @@ const LOOSE = new WeakSet<object>()
 // Converted schemas with an `anyOf` of two branches, for `mergeNullable`. It
 // runs once the whole input is converted, since a cycle can meet a branch
 // before its conversion is done.
-const twoBranchAnyOfs: Record<string, unknown>[] = []
+const twoBranchAnyOfs: [schema: Record<string, unknown>, input: object][] = []
+
+// Input schemas whose `anyOf` a `$ref` kept as written points into. They keep
+// it, so the `$ref` still resolves.
+const anyOfTargets = new Set<object>()
 
 // A branch with these stays: null can still fail them beside `nullable: true`,
 // and a `nullable` already there was written 3.0-style.
@@ -275,6 +279,12 @@ function refOr(convert: Convert): Convert {
 
 function convertSchemaRef(ref: string, ctx: Context): unknown {
   if (!isInlined(ref, ctx)) {
+    for (let at = ref.indexOf('/anyOf/'); at !== -1; at = ref.indexOf('/anyOf/', at + 1)) {
+      const holder = resolvePointer(ref.slice(0, at), ctx)
+      if (isRecord(holder)) {
+        anyOfTargets.add(holder)
+      }
+    }
     // A `$ref` kept as written loses what its target loses. Converting the
     // target here reuses the copy the components get, and stops at a cycle.
     const out = { $ref: ref }
@@ -661,7 +671,7 @@ function finishSchema(
     }
   }
   if (Array.isArray(out.anyOf) && out.anyOf.length === 2) {
-    twoBranchAnyOfs.push(out)
+    twoBranchAnyOfs.push([out, schema])
   }
   if (loose) {
     LOOSE.add(out)
@@ -788,10 +798,14 @@ function convertDocument(value: unknown, ctx: Context): unknown {
 
 function downgradeV30(root: unknown, convert: Convert): unknown {
   twoBranchAnyOfs.length = 0
+  anyOfTargets.clear()
   const out = downgrade(root, convert)
-  for (const schema of twoBranchAnyOfs.splice(0)) {
-    mergeNullable(schema, root)
+  for (const [schema, input] of twoBranchAnyOfs.splice(0)) {
+    if (!anyOfTargets.has(input)) {
+      mergeNullable(schema, root)
+    }
   }
+  anyOfTargets.clear()
   return out
 }
 
diff --git a/packages/downgrader/tests/v3.1-to-v3.0.test.ts b/packages/downgrader/tests/v3.1-to-v3.0.test.ts
index c895db6..d449cc3 100644
--- a/packages/downgrader/tests/v3.1-to-v3.0.test.ts
+++ b/packages/downgrader/tests/v3.1-to-v3.0.test.ts
@@ -844,6 +844,40 @@ describe('downgradeSchemaV31ToV30', () => {
       ).toEqual({ properties: { a: { type: 'string', nullable: true } } })
     })
 
+    it('keeps an anyOf of a branch and null that a $ref points into, so the $ref still resolves', () => {
+      const address = { type: 'object', properties: { city: { type: 'string' } } } as const
+      expect(
+        downgradeSchemaV31ToV30({
+          type: 'object',
+          properties: {
+            home: { anyOf: [address, { type: 'null' }] },
+            work: { anyOf: [{ $ref: '#/properties/home/anyOf/0' }, { type: 'null' }] },
+          },
+        }),
+      ).toEqual({
+        type: 'object',
+        properties: {
+          home: { anyOf: [address, { type: 'string', nullable: true, enum: [null] }] },
+          work: {
+            oneOf: [
+              { $ref: '#/properties/home/anyOf/0' },
+              { type: 'string', nullable: true, enum: [null] },
+            ],
+          },
+        },
+      })
+    })
+
+    it('keeps an anyOf of a branch and null that a $ref inside it points into', () => {
+      const node = {
+        type: 'object',
+        properties: { children: { type: 'array', items: { $ref: '#/anyOf/0' } } },
+      } as const
+      const out = downgradeSchemaV31ToV30({ anyOf: [node, { type: 'null' }] })
+      expect(out).toEqual({ anyOf: [node, { type: 'string', nullable: true, enum: [null] }] })
+      expect(() => JSON.stringify(out)).not.toThrow()
+    })
+
     it('leaves a $ref into $defs that does not resolve as written', () => {
       expect(downgradeSchemaV31ToV30({ $ref: '#/$defs/Missing' })).toEqual({
         $ref: '#/$defs/Missing',