diff --git a/clang/include/clang/Tooling/Syntax/Pseudo/Token.h b/clang/include/clang/Tooling/Syntax/Pseudo/Token.h
--- a/clang/include/clang/Tooling/Syntax/Pseudo/Token.h
+++ b/clang/include/clang/Tooling/Syntax/Pseudo/Token.h
@@ -195,6 +195,11 @@
 /// (And having cooked token kinds in PP-disabled sections is useful for us).
 TokenStream cook(const TokenStream &, const clang::LangOptions &);
 
+/// Derives a token stream by splitting the greatergreater token.
+///
+/// The greatergreater >> token is split into two greater tokens.
+TokenStream splitGreaterGreater(const TokenStream &);
+
 /// Drops comment tokens.
 TokenStream stripComments(const TokenStream &);
 
diff --git a/clang/lib/Tooling/Syntax/Pseudo/Token.cpp b/clang/lib/Tooling/Syntax/Pseudo/Token.cpp
--- a/clang/lib/Tooling/Syntax/Pseudo/Token.cpp
+++ b/clang/lib/Tooling/Syntax/Pseudo/Token.cpp
@@ -7,6 +7,7 @@
 //===----------------------------------------------------------------------===//
 
 #include "clang/Tooling/Syntax/Pseudo/Token.h"
+#include "clang/Basic/TokenKinds.h"
 #include "llvm/ADT/StringExtras.h"
 #include "llvm/Support/Format.h"
 #include "llvm/Support/FormatVariadic.h"
@@ -93,6 +94,29 @@
     OS << '\n';
 }
 
+TokenStream splitGreaterGreater(const TokenStream &Input) {
+  TokenStream Out;
+  for (Token T : Input.tokens()) {
+    // FIXME: split lessless token to support Cuda's triple angle brackets <<<.
+    if (T.Kind == tok::greatergreater) {
+      if (T.text() == ">>") {
+        T.Kind = tok::greater;
+        T.Length = 1;
+        Out.push(T);
+        T.Data = T.text().data() + 1;
+        // FIXME: Line is wrong if the first greater is followed by an escaped
+        // newline.
+        Out.push(T);
+        continue;
+      }
+      assert(false && "split an uncook token stream!");
+    }
+    Out.push(std::move(T));
+  }
+  Out.finalize();
+  return Out;
+}
+
 TokenStream stripComments(const TokenStream &Input) {
   TokenStream Out;
   for (const Token &T : Input.tokens()) {
diff --git a/clang/lib/Tooling/Syntax/Pseudo/cxx.bnf b/clang/lib/Tooling/Syntax/Pseudo/cxx.bnf
--- a/clang/lib/Tooling/Syntax/Pseudo/cxx.bnf
+++ b/clang/lib/Tooling/Syntax/Pseudo/cxx.bnf
@@ -13,6 +13,9 @@
 #  - the file merely describes the core C++ grammar. Preprocessor directives and
 #    lexical conversions are omitted as we reuse clang's lexer and run a fake
 #    preprocessor;
+#  - grammar rules with the >> token are adjusted, the greatergreater token is
+#    split into two > tokens, to make the GLR parser aware of nested templates
+#    and right shift operator.
 #
 # Guidelines:
 #   - non-terminals are lower_case; terminals (aka tokens) correspond to
@@ -96,7 +99,7 @@
 fold-operator := ^
 fold-operator := |
 fold-operator := <<
-fold-operator := >>
+fold-operator := greatergreater
 fold-operator := +=
 fold-operator := -=
 fold-operator := *=
@@ -202,7 +205,7 @@
 # expr.shift
 shift-expression := additive-expression
 shift-expression := shift-expression << additive-expression
-shift-expression := shift-expression >> additive-expression
+shift-expression := shift-expression greatergreater additive-expression
 # expr.spaceship
 compare-expression := shift-expression
 compare-expression := compare-expression <=> shift-expression
@@ -615,7 +618,7 @@
 operator-name := ^^
 operator-name := ||
 operator-name := <<
-operator-name := >>
+operator-name := greatergreater
 operator-name := <<=
 operator-name := >>=
 operator-name := ++
@@ -737,3 +740,8 @@
 module-keyword := IDENTIFIER
 import-keyword := IDENTIFIER
 export-keyword := IDENTIFIER
+
+#! Greatergreater token -- clang lexer always lexes it as a single token, we
+#! split it into two tokens to make the GLR parser aware of the nested-template
+#! case.
+greatergreater := > >
diff --git a/clang/unittests/Tooling/Syntax/Pseudo/TokenTest.cpp b/clang/unittests/Tooling/Syntax/Pseudo/TokenTest.cpp
--- a/clang/unittests/Tooling/Syntax/Pseudo/TokenTest.cpp
+++ b/clang/unittests/Tooling/Syntax/Pseudo/TokenTest.cpp
@@ -172,6 +172,30 @@
                             }));
 }
 
+TEST(TokenTest, SplitGreaterGreater) {
+  LangOptions Opts;
+  std::string Code = R"cpp(
+>> // split
+// >> with an escaped newline in the middle, split
+>\
+>
+>>= // not split
+)cpp";
+  TokenStream Raw = stripComments(cook(lex(Code, Opts), Opts));
+  EXPECT_THAT(Raw.tokens(),
+              ElementsAreArray({token(">>", tok::greatergreater),
+                                token(">>", tok::greatergreater),
+                                token(">>=", tok::greatergreaterequal)}));
+  TokenStream Split = splitGreaterGreater(Raw);
+  EXPECT_THAT(Split.tokens(), ElementsAreArray({
+                                  token(">", tok::greater),
+                                  token(">", tok::greater),
+                                  token(">", tok::greater),
+                                  token(">", tok::greater),
+                                  token(">>=", tok::greatergreaterequal),
+                              }));
+}
+
 TEST(TokenTest, DropComments) {
   LangOptions Opts;
   std::string Code = R"cpp(