diff --git a/llvm/include/llvm/DebugInfo/DWARF/DWARFAcceleratorTable.h b/llvm/include/llvm/DebugInfo/DWARF/DWARFAcceleratorTable.h
--- a/llvm/include/llvm/DebugInfo/DWARF/DWARFAcceleratorTable.h
+++ b/llvm/include/llvm/DebugInfo/DWARF/DWARFAcceleratorTable.h
@@ -743,6 +743,12 @@
   const NameIndex *getCUNameIndex(uint64_t CUOffset);
 };
 
+/// If `Name` is the name of a templated function that includes template
+/// parameters, returns a substring of `Name` containing no template
+/// parameters.
+/// E.g.: StripTemplateParameters("foo<int>") = "foo".
+std::optional<StringRef> StripTemplateParameters(StringRef Name);
+
 } // end namespace llvm
 
 #endif // LLVM_DEBUGINFO_DWARF_DWARFACCELERATORTABLE_H
diff --git a/llvm/lib/DWARFLinker/DWARFLinker.cpp b/llvm/lib/DWARFLinker/DWARFLinker.cpp
--- a/llvm/lib/DWARFLinker/DWARFLinker.cpp
+++ b/llvm/lib/DWARFLinker/DWARFLinker.cpp
@@ -15,6 +15,7 @@
 #include "llvm/DWARFLinker/DWARFLinkerDeclContext.h"
 #include "llvm/DWARFLinker/DWARFStreamer.h"
 #include "llvm/DebugInfo/DWARF/DWARFAbbreviationDeclaration.h"
+#include "llvm/DebugInfo/DWARF/DWARFAcceleratorTable.h"
 #include "llvm/DebugInfo/DWARF/DWARFContext.h"
 #include "llvm/DebugInfo/DWARF/DWARFDataExtractor.h"
 #include "llvm/DebugInfo/DWARF/DWARFDebugLine.h"
@@ -140,37 +141,6 @@
 
 DwarfEmitter::~DwarfEmitter() = default;
 
-static std::optional<StringRef> StripTemplateParameters(StringRef Name) {
-  // We are looking for template parameters to strip from Name. e.g.
-  //
-  //  operator<<B>
-  //
-  // We look for > at the end but if it does not contain any < then we
-  // have something like operator>>. We check for the operator<=> case.
-  if (!Name.endswith(">") || Name.count("<") == 0 || Name.endswith("<=>"))
-    return {};
-
-  // How many < until we have the start of the template parameters.
-  size_t NumLeftAnglesToSkip = 1;
-
-  // If we have operator<=> then we need to skip its < as well.
-  NumLeftAnglesToSkip += Name.count("<=>");
-
-  size_t RightAngleCount = Name.count('>');
-  size_t LeftAngleCount = Name.count('<');
-
-  // If we have more < than > we have operator< or operator<<
-  // we to account for their < as well.
-  if (LeftAngleCount > RightAngleCount)
-    NumLeftAnglesToSkip += LeftAngleCount - RightAngleCount;
-
-  size_t StartOfTemplate = 0;
-  while (NumLeftAnglesToSkip--)
-    StartOfTemplate = Name.find('<', StartOfTemplate) + 1;
-
-  return Name.substr(0, StartOfTemplate - 1);
-}
-
 bool DWARFLinker::DIECloner::getDIENames(const DWARFDie &Die,
                                          AttributesInfo &Info,
                                          OffsetsStringPool &StringPool,
diff --git a/llvm/lib/DebugInfo/DWARF/DWARFAcceleratorTable.cpp b/llvm/lib/DebugInfo/DWARF/DWARFAcceleratorTable.cpp
--- a/llvm/lib/DebugInfo/DWARF/DWARFAcceleratorTable.cpp
+++ b/llvm/lib/DebugInfo/DWARF/DWARFAcceleratorTable.cpp
@@ -969,3 +969,34 @@
   }
   return CUToNameIndex.lookup(CUOffset);
 }
+
+std::optional<StringRef> llvm::StripTemplateParameters(StringRef Name) {
+  // We are looking for template parameters to strip from Name. e.g.
+  //
+  //  operator<<B>
+  //
+  // We look for > at the end but if it does not contain any < then we
+  // have something like operator>>. We check for the operator<=> case.
+  if (!Name.endswith(">") || Name.count("<") == 0 || Name.endswith("<=>"))
+    return {};
+
+  // How many < until we have the start of the template parameters.
+  size_t NumLeftAnglesToSkip = 1;
+
+  // If we have operator<=> then we need to skip its < as well.
+  NumLeftAnglesToSkip += Name.count("<=>");
+
+  size_t RightAngleCount = Name.count('>');
+  size_t LeftAngleCount = Name.count('<');
+
+  // If we have more < than > we have operator< or operator<<
+  // we to account for their < as well.
+  if (LeftAngleCount > RightAngleCount)
+    NumLeftAnglesToSkip += LeftAngleCount - RightAngleCount;
+
+  size_t StartOfTemplate = 0;
+  while (NumLeftAnglesToSkip--)
+    StartOfTemplate = Name.find('<', StartOfTemplate) + 1;
+
+  return Name.substr(0, StartOfTemplate - 1);
+}
diff --git a/llvm/lib/DebugInfo/DWARF/DWARFVerifier.cpp b/llvm/lib/DebugInfo/DWARF/DWARFVerifier.cpp
--- a/llvm/lib/DebugInfo/DWARF/DWARFVerifier.cpp
+++ b/llvm/lib/DebugInfo/DWARF/DWARFVerifier.cpp
@@ -1351,12 +1351,18 @@
   return NumErrors;
 }
 
-static SmallVector<StringRef, 2> getNames(const DWARFDie &DIE,
+static SmallVector<StringRef, 3> getNames(const DWARFDie &DIE,
+                                          bool IncludeStrippedTemplateNames,
                                           bool IncludeLinkageName = true) {
   SmallVector<StringRef, 2> Result;
-  if (const char *Str = DIE.getShortName())
+  if (const char *Str = DIE.getShortName()) {
     Result.emplace_back(Str);
-  else if (DIE.getTag() == dwarf::DW_TAG_namespace)
+    if (IncludeStrippedTemplateNames) {
+      if (std::optional<StringRef> StrippedName =
+              StripTemplateParameters(Result.back()))
+        Result.push_back(*StrippedName);
+    }
+  } else if (DIE.getTag() == dwarf::DW_TAG_namespace)
     Result.emplace_back("(anonymous namespace)");
 
   if (IncludeLinkageName) {
@@ -1423,7 +1429,12 @@
       ++NumErrors;
     }
 
-    auto EntryNames = getNames(DIE);
+    // We allow an extra name for functions: their name without any template
+    // parameters.
+    auto IncludeStrippedTemplateNames =
+        DIE.getTag() == DW_TAG_subprogram ||
+        DIE.getTag() == DW_TAG_inlined_subroutine;
+    auto EntryNames = getNames(DIE, IncludeStrippedTemplateNames);
     if (!is_contained(EntryNames, Str)) {
       error() << formatv("Name Index @ {0:x}: Entry @ {1:x}: mismatched Name "
                          "of DIE @ {2:x}: index - {3}; debug_info - {4}.\n",
@@ -1496,7 +1507,11 @@
   // the linkage name."
   auto IncludeLinkageName = Die.getTag() == DW_TAG_subprogram ||
                             Die.getTag() == DW_TAG_inlined_subroutine;
-  auto EntryNames = getNames(Die, IncludeLinkageName);
+  // We *allow* stripped template names as an extra entry into the template,
+  // but we don't *require* them to pass the completeness test.
+  auto IncludeStrippedTemplateNames = false;
+  auto EntryNames =
+      getNames(Die, IncludeStrippedTemplateNames, IncludeLinkageName);
   if (EntryNames.empty())
     return 0;
 
diff --git a/llvm/test/tools/dsymutil/X86/dwarf5-accel.test b/llvm/test/tools/dsymutil/X86/dwarf5-accel.test
--- a/llvm/test/tools/dsymutil/X86/dwarf5-accel.test
+++ b/llvm/test/tools/dsymutil/X86/dwarf5-accel.test
@@ -13,12 +13,10 @@
 ## $ clang -gdwarf-5 dwarf5-accel.cpp -c -o dwarf5-accel.o
 
 #RUN: dsymutil -accelerator=Dwarf -oso-prepend-path %p/Inputs -y %s -o %t.dSYM
-#COM: Uncomment next line when dwarfdump will handle stripped template names.
-#COM: llvm-dwarfdump --verify  %t.dSYM | FileCheck %s --check-prefix VERIFY
+#RUN: llvm-dwarfdump --verify  %t.dSYM | FileCheck %s --check-prefix VERIFY
 #RUN: llvm-dwarfdump -a --verbose  %t.dSYM | FileCheck %s
 
-#COM: Uncomment next line when dwarfdump will handle stripped template names.
-#COM: #VERIFY: No errors.
+#VERIFY: No errors.
 
 #CHECK: .debug_names
 #CHECK: "foo"