diff --git a/Code/Mantid/Framework/LiveData/src/MonitorLiveData.cpp b/Code/Mantid/Framework/LiveData/src/MonitorLiveData.cpp
index 7f24610b6c42..6984e4984f8e 100644
--- a/Code/Mantid/Framework/LiveData/src/MonitorLiveData.cpp
+++ b/Code/Mantid/Framework/LiveData/src/MonitorLiveData.cpp
@@ -57,12 +57,16 @@ namespace LiveData
   }
 
   //----------------------------------------------------------------------------------------------
-  /** Clone a workspace, if there is enough memory available */
+  /** Clone a workspace, if there is enough memory available.
+      It needs to be a clone rather than a rename so that an open instrument window continues
+      to track the live workspace.
+   */
   void MonitorLiveData::doClone(const std::string & originalName, const std::string & newName)
   {
-    if (AnalysisDataService::Instance().doesExist(originalName))
+    auto & ads = AnalysisDataService::Instance();
+    if ( ads.doesExist(originalName) )
     {
-      Workspace_sptr original = AnalysisDataService::Instance().retrieveWS<Workspace>(originalName);
+      Workspace_sptr original = ads.retrieveWS<Workspace>(originalName);
       if (original)
       {
         size_t bytesUsed = original->getMemorySize();
@@ -71,11 +75,29 @@ namespace LiveData
         if (size_t(3)*bytesUsed < bytesAvail)
         {
           WriteLock _lock(*original);
+
+          // Clone the monitor workspace, if there is one
+          auto originalMatrix = boost::dynamic_pointer_cast<MatrixWorkspace>(original);
+          MatrixWorkspace_sptr monitorWS, newMonitorWS;
+          if ( originalMatrix && ( monitorWS = originalMatrix->monitorWorkspace() ) )
+          {
+            auto monitorsCloner = createChildAlgorithm("CloneWorkspace", 0, 0, false);
+            monitorsCloner->setProperty("InputWorkspace", monitorWS);
+            monitorsCloner->executeAsChildAlg();
+            Workspace_sptr outputWS = monitorsCloner->getProperty("OutputWorkspace");
+            newMonitorWS = boost::dynamic_pointer_cast<MatrixWorkspace>(outputWS);
+          }
+
           Algorithm_sptr cloner = createChildAlgorithm("CloneWorkspace", 0, 0, false);
           cloner->setPropertyValue("InputWorkspace", originalName);
           cloner->setPropertyValue("OutputWorkspace", newName);
           cloner->setAlwaysStoreInADS(true); // We must force the ADS to be updated
           cloner->executeAsChildAlg();
+
+          if ( newMonitorWS ) // If there was a monitor workspace, set it back on the result
+          {
+            ads.retrieveWS<MatrixWorkspace>(newName)->setMonitorWorkspace(newMonitorWS);
+          }
         }
         else
         {
diff --git a/Code/Mantid/Framework/LiveData/test/MonitorLiveDataTest.h b/Code/Mantid/Framework/LiveData/test/MonitorLiveDataTest.h
index 1595db78fb6f..c9b558464298 100644
--- a/Code/Mantid/Framework/LiveData/test/MonitorLiveDataTest.h
+++ b/Code/Mantid/Framework/LiveData/test/MonitorLiveDataTest.h
@@ -211,10 +211,16 @@ class MonitorLiveDataTest : public CxxTest::TestSuite
     // The first workspace got cloned to a new name (the suffix is set in the TestDataListener)
     EventWorkspace_sptr ws1 = AnalysisDataService::Instance().retrieveWS<EventWorkspace>("fake2_999");
     TS_ASSERT_EQUALS( ws1->getNumberEvents(), 4*200);
+    // Make sure the monitor workspace is present and correct
+    TS_ASSERT( ws1->monitorWorkspace() );
+    TS_ASSERT_EQUALS( ws1->monitorWorkspace()->dataY(0)[0], 4 );
 
     // And this is the current run
     EventWorkspace_sptr ws2 = AnalysisDataService::Instance().retrieveWS<EventWorkspace>("fake2");
     TS_ASSERT_EQUALS( ws2->getNumberEvents(), 200);
+    // Make sure the monitor workspace is present and correct
+    TS_ASSERT( ws2->monitorWorkspace() );
+    TS_ASSERT_EQUALS( ws2->monitorWorkspace()->dataY(0)[0], 1 );
 
     Kernel::Timer timer;
     while ( alg1->isRunning() && timer.elapsed_no_reset() < 0.5 ) {}
diff --git a/Code/Mantid/Framework/LiveData/test/TestDataListener.cpp b/Code/Mantid/Framework/LiveData/test/TestDataListener.cpp
index c59e5cb40b3b..f97dd79855c8 100644
--- a/Code/Mantid/Framework/LiveData/test/TestDataListener.cpp
+++ b/Code/Mantid/Framework/LiveData/test/TestDataListener.cpp
@@ -92,6 +92,9 @@ namespace LiveData
     m_buffer->setInstrument(inst);
     // Set a run number
     m_buffer->mutableRun().addProperty("run_number", std::string("999"));
+    // Add a monitor workspace
+    auto monitorWS = WorkspaceFactory::Instance().create(m_buffer);
+    m_buffer->setMonitorWorkspace(monitorWS);
   }
 
   boost::shared_ptr<Workspace> TestDataListener::extractData()
@@ -107,6 +110,7 @@ namespace LiveData
       el1.addEventQuickly(TofEvent(m_rand->nextValue()));
       el2.addEventQuickly(TofEvent(m_rand->nextValue()));
     }
+    m_buffer->monitorWorkspace()->dataY(0)[0] = 1.0;
 
     // Copy the workspace pointer to a temporary variable
     EventWorkspace_sptr extracted = m_buffer;