Try 2 - Change how ui::Clipboard is accessed so there's only one per thread.

ui/base/clipboard/clipboard.h

Index: ui/base/clipboard/clipboard.h
diff --git a/ui/base/clipboard/clipboard.h b/ui/base/clipboard/clipboard.h
index b98cfc4748953f32ad60eb5c45be673cd1afc771..18463da7472806a860d8faf7b3abc229c4933416 100644
--- a/ui/base/clipboard/clipboard.h
+++ b/ui/base/clipboard/clipboard.h
@@ -15,6 +15,7 @@
 #include "base/shared_memory.h"
 #include "base/string16.h"
 #include "base/threading/thread_checker.h"
+#include "base/threading/platform_thread.h"
 #include "ui/base/ui_export.h"
 
 #if defined(TOOLKIT_GTK)
@@ -46,6 +47,7 @@ class NSString;
 #endif
 
 namespace ui {
+class ClipboardTest;
 
 class UI_EXPORT Clipboard : NON_EXPORTED_BASE(public base::ThreadChecker) {
  public:
@@ -182,8 +184,22 @@ class UI_EXPORT Clipboard : NON_EXPORTED_BASE(public base::ThreadChecker) {
     return static_cast<Buffer>(buffer);
   }
 
-  Clipboard();
-  ~Clipboard();
+  // Sets the list of threads that are allowed to access the clipboard.
+  static void SetAllowedThreads(
+      const std::vector<base::PlatformThreadId>& allowed_threads);
+
+  // Returns the clipboard object for the current thread.
+  //
+  // Most implementations will have at most one clipboard which will live on
+  // the main UI thread, but Windows has tricky semantics where there have to
+  // be two clipboards: one that lives on the UI thread and one that lives on
+  // the IO thread.
+  static Clipboard* GetForCurrentThread();
+
+  // Destroys the clipboard for the current thread. Usually, this will clean up
+  // all clipboards, except on Windows. (Previous code leaks the IO thread
+  // clipboard, so it shouldn't be a problem.)
+  static void DestroyClipboardForCurrentThread();
 
   // Write a bunch of objects to the system clipboard. Copies are made of the
   // contents of |objects|. On Windows they are copied to the system clipboard.
@@ -284,6 +300,10 @@ class UI_EXPORT Clipboard : NON_EXPORTED_BASE(public base::ThreadChecker) {
  private:
   FRIEND_TEST_ALL_PREFIXES(ClipboardTest, SharedBitmapTest);
   FRIEND_TEST_ALL_PREFIXES(ClipboardTest, EmptyHTMLTest);
+  friend class ClipboardTest;
+
+  Clipboard();
+  ~Clipboard();
 
   void DispatchObject(ObjectType type, const ObjectMapParams& params);