diff --git a/llvm/lib/Transforms/IPO/FunctionSpecialization.cpp b/llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
--- a/llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
+++ b/llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
@@ -53,6 +53,7 @@
 #include "llvm/Analysis/TargetTransformInfo.h"
 #include "llvm/Analysis/ValueLattice.h"
 #include "llvm/Analysis/ValueLatticeUtils.h"
+#include "llvm/Analysis/ValueTracking.h"
 #include "llvm/IR/IntrinsicInst.h"
 #include "llvm/Transforms/Scalar/SCCP.h"
 #include "llvm/Transforms/Utils/Cloning.h"
@@ -732,22 +733,19 @@
   if (isa<PoisonValue>(V))
     return nullptr;
 
-  // TrackValueOfGlobalVariable only tracks scalar global variables.
-  if (auto *GV = dyn_cast<GlobalVariable>(V)) {
-    // Check if we want to specialize on the address of non-constant
-    // global values.
-    if (!GV->isConstant() && !SpecializeOnAddress)
-      return nullptr;
-
-    if (!GV->getValueType()->isSingleValueType())
-      return nullptr;
-  }
-
   // Select for possible specialisation values that are constants or
   // are deduced to be constants or constant ranges with a single element.
   Constant *C = dyn_cast<Constant>(V);
   if (!C)
     C = Solver.getConstantOrNull(V);
+
+  // Don't specialize on (anything derived from) the address of a non-constant
+  // global variable, unless explicitly enabled.
+  if (C && C->getType()->isPointerTy() && !C->isNullValue())
+    if (auto *GV = dyn_cast<GlobalVariable>(getUnderlyingObject(C));
+        GV && !(GV->isConstant() || SpecializeOnAddress))
+      return nullptr;
+
   return C;
 }
 
diff --git a/llvm/test/Transforms/FunctionSpecialization/compiler-crash-promote-alloca.ll b/llvm/test/Transforms/FunctionSpecialization/compiler-crash-promote-alloca.ll
--- a/llvm/test/Transforms/FunctionSpecialization/compiler-crash-promote-alloca.ll
+++ b/llvm/test/Transforms/FunctionSpecialization/compiler-crash-promote-alloca.ll
@@ -4,7 +4,7 @@
 ; while attempting to promote the alloca in `entry`.
 ; CHECK: bar.1
 
-@block = internal global [8 x i8] zeroinitializer, align 1
+@block = internal constant [8 x i8] zeroinitializer, align 1
 
 define dso_local void @entry() {
   %1 = alloca i32, align 4
diff --git a/llvm/test/Transforms/FunctionSpecialization/global-var-constants.ll b/llvm/test/Transforms/FunctionSpecialization/global-var-constants.ll
new file mode 100644
--- /dev/null
+++ b/llvm/test/Transforms/FunctionSpecialization/global-var-constants.ll
@@ -0,0 +1,78 @@
+; RUN:  opt -S --passes=ipsccp,deadargelim --force-specialization                       < %s | FileCheck %s --check-prefix=NO-GLOBALS
+; RUN:  opt -S --passes=ipsccp,deadargelim --force-specialization --funcspec-on-address < %s | FileCheck %s --check-prefix=GLOBALS
+@G = global [10 x i32] zeroinitializer, align 4
+
+define internal i32 @f(ptr %p) noinline {
+entry:
+  %0 = load i32, ptr %p, align 4
+  store i32 0, ptr %p, align 4
+  ret i32 %0
+}
+
+define internal i32 @g(i32 %x, i32 %y, ptr %p) noinline {
+entry:
+  %cmp = icmp sgt i32 %x, %y
+  br i1 %cmp, label %if.then, label %if.else
+
+if.then:
+  br label %if.end
+
+if.else:
+  br label %if.end
+
+if.end:
+  %x.addr.0 = phi i32 [ %x, %if.then ], [ 11, %if.else ]
+  %p.addr.0 = phi ptr [ @G, %if.then ], [ %p, %if.else ]
+  %call = call i32 @f(ptr %p.addr.0)
+  %add = add nsw i32 %call, %x.addr.0
+  ret i32 %add
+}
+
+define i32 @h0(ptr %p) {
+entry:
+  %call = call i32 @g(i32 2, i32 1, ptr %p)
+  ret i32 %call
+}
+
+define i32 @h1() {
+entry:
+  %call = call i32 @f(ptr @G)
+  ret i32 %call
+}
+
+define i32 @h2() {
+entry:
+  %call = call i32 @f(ptr getelementptr inbounds (i32, ptr @G, i64 1))
+  ret i32 %call
+}
+
+; Check if specialisation on the address of a non-const global variable
+; is not allowed, then it is not performed.
+
+; NO-GLOBALS-LABEL: define internal i32 @g()
+; NO-GLOBALS: call i32 @f(ptr @G)
+
+; NO-GLOBALS-LABEL: define i32 @h0(ptr %p)
+; NO-GLOBALS:call i32 @g()
+
+; NO-GLOBALS-LABEL: define i32 @h1()
+; NO-GLOBALS: call i32 @f(ptr @G)
+
+; NO-GLOBALS-LABEL: define i32 @h2()
+; NO-GLOBALS: call i32 @f(ptr getelementptr inbounds (i32, ptr @G, i64 1))
+
+; Check if specialisation on the address of a non-const global variable
+; is allowed, then it is performed where possible.
+
+; GLOBALS-LABEL: define internal i32 @g()
+; GLOBALS: call i32 @f.2()
+
+; GLOBALS-LABEL: define i32 @h0(ptr %p)
+; GLOBALS: call i32 @g()
+
+; GLOBALS-LABEL: define i32 @h1()
+; GLOBALS: call i32 @f.2()
+
+; GLOBALS-LABEL: define i32 @h2()
+; GLOBALS: call i32 @f.1()
+