diff --git a/config/add-overlay-annotations.py b/config/add-overlay-annotations.py
index 85b42026d8d7..0a30eee5799b 100644
--- a/config/add-overlay-annotations.py
+++ b/config/add-overlay-annotations.py
@@ -177,6 +177,12 @@ def insert_overlay_caller_annotations(lines):
         out_lines.append(line)
     return out_lines
 
+explicitly_global = set([
+    "java/ql/lib/semmle/code/java/dispatch/VirtualDispatch.qll",
+    "java/ql/lib/semmle/code/java/dispatch/DispatchFlow.qll",
+    "java/ql/lib/semmle/code/java/dispatch/ObjFlow.qll",
+    "java/ql/lib/semmle/code/java/dispatch/internal/Unification.qll",
+])
 
 def annotate_as_appropriate(filename, lines):
     '''
@@ -196,6 +202,9 @@ def annotate_as_appropriate(filename, lines):
         ((filename.endswith("Query.qll") or filename.endswith("Config.qll")) and
          any("implements DataFlow::ConfigSig" in line for line in lines))):
         return None
+    elif filename in explicitly_global:
+        # These files are explicitly global and should not be annotated.
+        return None
     elif not any(line for line in lines if line.strip()):
         return None
 
diff --git a/java/ql/lib/semmle/code/java/dataflow/SSA.qll b/java/ql/lib/semmle/code/java/dataflow/SSA.qll
index dd902b70e35a..962f38e80ea6 100644
--- a/java/ql/lib/semmle/code/java/dataflow/SSA.qll
+++ b/java/ql/lib/semmle/code/java/dataflow/SSA.qll
@@ -253,17 +253,18 @@ class SsaImplicitUpdate extends SsaUpdate {
     or
     if this.hasImplicitQualifierUpdate()
     then
-      if exists(this.getANonLocalUpdate())
+      if isNonLocal(this)
       then result = "nonlocal + nonlocal qualifier"
       else result = "nonlocal qualifier"
     else (
-      exists(this.getANonLocalUpdate()) and result = "nonlocal"
+      isNonLocal(this) and result = "nonlocal"
     )
   }
 
   /**
    * Gets a reachable `FieldWrite` that might represent this ssa update, if any.
    */
+  overlay[global]
   FieldWrite getANonLocalUpdate() {
     exists(SsaSourceField f, Callable setter |
       relevantFieldUpdate(setter, f.getField(), result) and
@@ -287,6 +288,11 @@ class SsaImplicitUpdate extends SsaUpdate {
   }
 }
 
+overlay[global]
+private predicate isNonLocalImpl(SsaImplicitUpdate su) { exists(su.getANonLocalUpdate()) }
+
+private predicate isNonLocal(SsaImplicitUpdate su) = forceLocal(isNonLocalImpl/1)(su)
+
 /**
  * An SSA variable that represents an uncertain implicit update of the value.
  * This is a `Call` that might reach a non-local update of the field or one of
diff --git a/java/ql/lib/semmle/code/java/dataflow/internal/SsaImpl.qll b/java/ql/lib/semmle/code/java/dataflow/internal/SsaImpl.qll
index dc2cb7e7d00d..275a0afafc08 100644
--- a/java/ql/lib/semmle/code/java/dataflow/internal/SsaImpl.qll
+++ b/java/ql/lib/semmle/code/java/dataflow/internal/SsaImpl.qll
@@ -157,15 +157,20 @@ private predicate hasEntryDef(TrackedVar v, BasicBlock b) {
 }
 
 /** Holds if `n` might update the locally tracked variable `v`. */
+overlay[global]
 pragma[nomagic]
-private predicate uncertainVariableUpdate(TrackedVar v, ControlFlowNode n, BasicBlock b, int i) {
+private predicate uncertainVariableUpdateImpl(TrackedVar v, ControlFlowNode n, BasicBlock b, int i) {
   exists(Call c | c = n.asCall() | updatesNamedField(c, v, _)) and
   b.getNode(i) = n and
   hasDominanceInformation(b)
   or
-  uncertainVariableUpdate(v.getQualifier(), n, b, i)
+  uncertainVariableUpdateImpl(v.getQualifier(), n, b, i)
 }
 
+/** Holds if `n` might update the locally tracked variable `v`. */
+predicate uncertainVariableUpdate(TrackedVar v, ControlFlowNode n, BasicBlock b, int i) =
+  forceLocal(uncertainVariableUpdateImpl/4)(v, n, b, i)
+
 private module SsaInput implements SsaImplCommon::InputSig<Location, BasicBlock> {
   class SourceVariable = SsaSourceVariable;
 
@@ -335,6 +340,7 @@ private module Cached {
    *   Constructor --(intraInstanceCallEdge)-->+ Method(setter of this.f)
    * ```
    */
+  overlay[global]
   private predicate intraInstanceCallEdge(Callable c1, Method m2) {
     exists(MethodCall ma, RefType t1 |
       ma.getCaller() = c1 and
@@ -355,6 +361,7 @@ private module Cached {
     )
   }
 
+  overlay[global]
   private Callable tgt(Call c) {
     result = viableImpl_v2(c)
     or
@@ -364,11 +371,13 @@ private module Cached {
   }
 
   /** Holds if `(c1,c2)` is an edge in the call graph. */
+  overlay[global]
   private predicate callEdge(Callable c1, Callable c2) {
     exists(Call c | c.getCaller() = c1 and c2 = tgt(c))
   }
 
   /** Holds if `(c1,c2)` is an edge in the call graph excluding `intraInstanceCallEdge`. */
+  overlay[global]
   private predicate crossInstanceCallEdge(Callable c1, Callable c2) {
     callEdge(c1, c2) and not intraInstanceCallEdge(c1, c2)
   }
@@ -382,6 +391,7 @@ private module Cached {
     relevantFieldUpdate(_, f.getField(), _)
   }
 
+  overlay[global]
   private predicate source(Call call, TrackedField f, Field field, Callable c, boolean fresh) {
     relevantCall(call, f) and
     field = f.getField() and
@@ -395,9 +405,11 @@ private module Cached {
    * `fresh` indicates whether the instance `this` in `c` has been freshly
    * allocated along the call-chain.
    */
+  overlay[global]
   private newtype TCallableNode =
     MkCallableNode(Callable c, boolean fresh) { source(_, _, _, c, fresh) or edge(_, c, fresh) }
 
+  overlay[global]
   private predicate edge(TCallableNode n, Callable c2, boolean f2) {
     exists(Callable c1, boolean f1 | n = MkCallableNode(c1, f1) |
       intraInstanceCallEdge(c1, c2) and f2 = f1
@@ -407,6 +419,7 @@ private module Cached {
     )
   }
 
+  overlay[global]
   private predicate edge(TCallableNode n1, TCallableNode n2) {
     exists(Callable c2, boolean f2 |
       edge(n1, c2, f2) and
@@ -414,6 +427,7 @@ private module Cached {
     )
   }
 
+  overlay[global]
   pragma[noinline]
   private predicate source(Call call, TrackedField f, Field field, TCallableNode n) {
     exists(Callable c, boolean fresh |
@@ -422,24 +436,28 @@ private module Cached {
     )
   }
 
+  overlay[global]
   private predicate sink(Callable c, Field f, TCallableNode n) {
     setsOwnField(c, f) and n = MkCallableNode(c, false)
     or
     setsOtherField(c, f) and n = MkCallableNode(c, _)
   }
 
+  overlay[global]
   private predicate prunedNode(TCallableNode n) {
     sink(_, _, n)
     or
     exists(TCallableNode mid | edge(n, mid) and prunedNode(mid))
   }
 
+  overlay[global]
   private predicate prunedEdge(TCallableNode n1, TCallableNode n2) {
     prunedNode(n1) and
     prunedNode(n2) and
     edge(n1, n2)
   }
 
+  overlay[global]
   private predicate edgePlus(TCallableNode c1, TCallableNode c2) = fastTC(prunedEdge/2)(c1, c2)
 
   /**
@@ -447,6 +465,7 @@ private module Cached {
    * where `f` and `call` share the same enclosing callable in which a
    * `FieldRead` of `f` is reachable from `call`.
    */
+  overlay[global]
   pragma[noopt]
   private predicate updatesNamedFieldImpl(Call call, TrackedField f, Callable setter) {
     exists(TCallableNode src, TCallableNode sink, Field field |
@@ -457,11 +476,13 @@ private module Cached {
   }
 
   bindingset[call, f]
+  overlay[global]
   pragma[inline_late]
   private predicate updatesNamedField0(Call call, TrackedField f, Callable setter) {
     updatesNamedField(call, f, setter)
   }
 
+  overlay[global]
   cached
   predicate defUpdatesNamedField(SsaImplicitUpdate def, TrackedField f, Callable setter) {
     f = def.getSourceVariable() and
diff --git a/java/ql/lib/semmle/code/java/dispatch/DispatchFlow.qll b/java/ql/lib/semmle/code/java/dispatch/DispatchFlow.qll
index a9988e920c62..bd293eed6b3a 100644
--- a/java/ql/lib/semmle/code/java/dispatch/DispatchFlow.qll
+++ b/java/ql/lib/semmle/code/java/dispatch/DispatchFlow.qll
@@ -5,8 +5,6 @@
  * data flow check for lambdas, anonymous classes, and other sufficiently
  * private classes where all object instantiations are accounted for.
  */
-overlay[local?]
-module;
 
 import java
 private import VirtualDispatch
diff --git a/java/ql/lib/semmle/code/java/dispatch/ObjFlow.qll b/java/ql/lib/semmle/code/java/dispatch/ObjFlow.qll
index 86915f802743..a3da9118acc7 100644
--- a/java/ql/lib/semmle/code/java/dispatch/ObjFlow.qll
+++ b/java/ql/lib/semmle/code/java/dispatch/ObjFlow.qll
@@ -6,8 +6,6 @@
  * The set of dispatch targets for `Object.toString()` calls are reduced based
  * on possible data flow from objects of more specific types to the qualifier.
  */
-overlay[local?]
-module;
 
 import java
 private import VirtualDispatch
diff --git a/java/ql/lib/semmle/code/java/dispatch/VirtualDispatch.qll b/java/ql/lib/semmle/code/java/dispatch/VirtualDispatch.qll
index 877a62fb9455..78bf1ad0bdc1 100644
--- a/java/ql/lib/semmle/code/java/dispatch/VirtualDispatch.qll
+++ b/java/ql/lib/semmle/code/java/dispatch/VirtualDispatch.qll
@@ -2,8 +2,6 @@
  * Provides predicates for reasoning about runtime call targets through virtual
  * dispatch.
  */
-overlay[local?]
-module;
 
 import java
 import semmle.code.java.dataflow.TypeFlow
diff --git a/java/ql/lib/semmle/code/java/dispatch/internal/Unification.qll b/java/ql/lib/semmle/code/java/dispatch/internal/Unification.qll
index cd585de58e4e..6c92f7298d92 100644
--- a/java/ql/lib/semmle/code/java/dispatch/internal/Unification.qll
+++ b/java/ql/lib/semmle/code/java/dispatch/internal/Unification.qll
@@ -1,8 +1,6 @@
 /**
  * Provides a module to check whether two `ParameterizedType`s are unifiable.
  */
-overlay[local?]
-module;
 
 import java