diff --git a/reference-implementation/lib/abstract-ops/readable-streams.js b/reference-implementation/lib/abstract-ops/readable-streams.js
index 843ced877..f3c790b50 100644
--- a/reference-implementation/lib/abstract-ops/readable-streams.js
+++ b/reference-implementation/lib/abstract-ops/readable-streams.js
@@ -141,7 +141,8 @@ function ReadableStreamPipeTo(source, dest, preventClose, preventAbort, preventC
 
   let shuttingDown = false;
 
-  // This is used to keep track of the spec's requirement that we wait for ongoing writes during shutdown.
+  // This is used to keep track of the spec's requirement that we wait for ongoing reads and writes during shutdown.
+  let currentRead = promiseResolvedWith(undefined);
   let currentWrite = promiseResolvedWith(undefined);
 
   return new Promise((resolve, reject) => {
@@ -200,14 +201,13 @@ function ReadableStreamPipeTo(source, dest, preventClose, preventAbort, preventC
       }
 
       return transformPromiseWith(writer._readyPromise, () => {
-        return new Promise((resolveRead, rejectRead) => {
+        currentRead = new Promise((resolveRead, rejectRead) => {
           ReadableStreamDefaultReaderRead(
             reader,
             {
               chunkSteps: chunk => {
-                currentWrite = transformPromiseWith(
-                  WritableStreamDefaultWriterWrite(writer, chunk), undefined, () => {}
-                );
+                currentWrite = WritableStreamDefaultWriterWrite(writer, chunk);
+                setPromiseIsHandledToTrue(currentWrite);
                 resolveRead(false);
               },
               closeSteps: () => resolveRead(true),
@@ -215,6 +215,7 @@ function ReadableStreamPipeTo(source, dest, preventClose, preventAbort, preventC
             }
           );
         });
+        return currentRead;
       });
     }
 
@@ -259,13 +260,38 @@ function ReadableStreamPipeTo(source, dest, preventClose, preventAbort, preventC
     setPromiseIsHandledToTrue(pipeLoop());
 
     function waitForWritesToFinish() {
-      // Another write may have started while we were waiting on this currentWrite, so we have to be sure to wait
-      // for that too.
-      const oldCurrentWrite = currentWrite;
-      return transformPromiseWith(
-        currentWrite,
-        () => oldCurrentWrite !== currentWrite ? waitForWritesToFinish() : undefined
-      );
+      let oldCurrentWrite;
+      return promiseResolvedWith(check());
+
+      function check() {
+        // Another write may have started while we were waiting on this currentWrite,
+        // so we have to be sure to wait for that too.
+        if (oldCurrentWrite !== currentWrite) {
+          oldCurrentWrite = currentWrite;
+          return transformPromiseWith(currentWrite, check, check);
+        }
+        return undefined;
+      }
+    }
+
+    function waitForReadsAndWritesToFinish() {
+      let oldCurrentRead;
+      let oldCurrentWrite;
+      return promiseResolvedWith(check());
+
+      function check() {
+        // Another read or write may have started while we were waiting on this currentRead or currentWrite,
+        // so we have to be sure to wait for that too.
+        if (oldCurrentRead !== currentRead) {
+          oldCurrentRead = currentRead;
+          return transformPromiseWith(currentRead, check, check);
+        }
+        if (oldCurrentWrite !== currentWrite) {
+          oldCurrentWrite = currentWrite;
+          return transformPromiseWith(currentWrite, check, check);
+        }
+        return undefined;
+      }
     }
 
     function isOrBecomesErrored(stream, promise, action) {
@@ -299,8 +325,8 @@ function ReadableStreamPipeTo(source, dest, preventClose, preventAbort, preventC
       function doTheRest() {
         uponPromise(
           action(),
-          () => finalize(originalIsError, originalError),
-          newError => finalize(true, newError)
+          () => waitForReadsAndWritesThenFinalize(originalIsError, originalError),
+          newError => waitForReadsAndWritesThenFinalize(true, newError)
         );
       }
     }
@@ -311,11 +337,11 @@ function ReadableStreamPipeTo(source, dest, preventClose, preventAbort, preventC
       }
       shuttingDown = true;
 
-      if (dest._state === 'writable' && WritableStreamCloseQueuedOrInFlight(dest) === false) {
-        uponFulfillment(waitForWritesToFinish(), () => finalize(isError, error));
-      } else {
-        finalize(isError, error);
-      }
+      waitForReadsAndWritesThenFinalize(isError, error);
+    }
+
+    function waitForReadsAndWritesThenFinalize(isError, error) {
+      uponFulfillment(waitForReadsAndWritesToFinish(), () => finalize(isError, error));
     }
 
     function finalize(isError, error) {