diff --git a/src/coreclr/tools/aot/ILCompiler.ReadyToRun/JitInterface/CorInfoImpl.ReadyToRun.cs b/src/coreclr/tools/aot/ILCompiler.ReadyToRun/JitInterface/CorInfoImpl.ReadyToRun.cs
index faf5ab5bd7b747..a7c65283f91893 100644
--- a/src/coreclr/tools/aot/ILCompiler.ReadyToRun/JitInterface/CorInfoImpl.ReadyToRun.cs
+++ b/src/coreclr/tools/aot/ILCompiler.ReadyToRun/JitInterface/CorInfoImpl.ReadyToRun.cs
@@ -573,6 +573,7 @@ public static bool ShouldCodeNotBeCompiledIntoFinalImage(InstructionSetSupport i
             var handle = ecmaMethod.Handle;
 
             List<TypeDesc> compExactlyDependsOnList = null;
+            bool hasCompHasFallback = false;
 
             foreach (var attributeHandle in metadataReader.GetMethodDefinition(handle).GetCustomAttributes())
             {
@@ -606,35 +607,41 @@ public static bool ShouldCodeNotBeCompiledIntoFinalImage(InstructionSetSupport i
                             compExactlyDependsOnList.Add(typeForBypass);
                         }
                     }
+                    else if (metadataReader.StringComparer.Equals(nameHandle, "CompHasFallbackAttribute"))
+                    {
+                        hasCompHasFallback = true;
+                    }
                 }
             }
 
             if (compExactlyDependsOnList != null && compExactlyDependsOnList.Count > 0)
             {
-                // Default to true, and set to false if at least one of the types is actually supported in the current environment, and none of the
-                // intrinsic types are in an opportunistic state.
-                bool doBypass = true;
+                bool anySupported = false;
 
                 foreach (var intrinsicType in compExactlyDependsOnList)
                 {
                     InstructionSet instructionSet = InstructionSetParser.LookupPlatformIntrinsicInstructionSet(intrinsicType.Context.Target.Architecture, intrinsicType);
-                    if (instructionSet == InstructionSet.ILLEGAL)
-                    {
-                        // This instruction set isn't supported on the current platform at all.
-                        continue;
-                    }
-                    if (instructionSetSupport.IsInstructionSetSupported(instructionSet) || instructionSetSupport.IsInstructionSetExplicitlyUnsupported(instructionSet))
-                    {
-                        doBypass = false;
-                    }
-                    else
+                    // If the instruction set is ILLEGAL, it means it is never supported by the current architecture so the behavior at runtime is known
+                    if (instructionSet != InstructionSet.ILLEGAL)
                     {
-                        // If we reach here this is an instruction set generally supported on this platform, but we don't know what the behavior will be at runtime
-                        return true;
+                        if (instructionSetSupport.IsInstructionSetSupported(instructionSet))
+                        {
+                            anySupported = true;
+                        }
+                        else if (!instructionSetSupport.IsInstructionSetExplicitlyUnsupported(instructionSet))
+                        {
+                            // If we reach here this is an instruction set generally supported on this platform, but we don't know what the behavior will be at runtime
+                            return true;
+                        }
                     }
                 }
 
-                return doBypass;
+                if (!anySupported && !hasCompHasFallback)
+                {
+                    // If none of the instruction sets are supported (all are either illegal or explicitly unsupported),
+                    // skip compilation unless the method has a functional fallback path
+                    return true;
+                }
             }
 
             // No reason to bypass compilation and code generation.
diff --git a/src/libraries/System.Private.CoreLib/src/System.Private.CoreLib.Shared.projitems b/src/libraries/System.Private.CoreLib/src/System.Private.CoreLib.Shared.projitems
index c744f85d3e05ce..6308788f0d0789 100644
--- a/src/libraries/System.Private.CoreLib/src/System.Private.CoreLib.Shared.projitems
+++ b/src/libraries/System.Private.CoreLib/src/System.Private.CoreLib.Shared.projitems
@@ -837,6 +837,7 @@
     <Compile Include="$(MSBuildThisFileDirectory)System\Runtime\CompilerServices\CallingConventions.cs" />
     <Compile Include="$(MSBuildThisFileDirectory)System\Runtime\CompilerServices\CollectionBuilderAttribute.cs" />
     <Compile Include="$(MSBuildThisFileDirectory)System\Runtime\CompilerServices\CompExactlyDependsOnAttribute.cs" />
+    <Compile Include="$(MSBuildThisFileDirectory)System\Runtime\CompilerServices\CompHasFallbackAttribute.cs" />
     <Compile Include="$(MSBuildThisFileDirectory)System\Runtime\CompilerServices\CompilationRelaxations.cs" />
     <Compile Include="$(MSBuildThisFileDirectory)System\Runtime\CompilerServices\CompilationRelaxationsAttribute.cs" />
     <Compile Include="$(MSBuildThisFileDirectory)System\Runtime\CompilerServices\CompilerFeatureRequiredAttribute.cs" />
diff --git a/src/libraries/System.Private.CoreLib/src/System/Runtime/CompilerServices/CompHasFallbackAttribute.cs b/src/libraries/System.Private.CoreLib/src/System/Runtime/CompilerServices/CompHasFallbackAttribute.cs
new file mode 100644
index 00000000000000..9ce5344c277d86
--- /dev/null
+++ b/src/libraries/System.Private.CoreLib/src/System/Runtime/CompilerServices/CompHasFallbackAttribute.cs
@@ -0,0 +1,18 @@
+// Licensed to the .NET Foundation under one or more agreements.
+// The .NET Foundation licenses this file to you under the MIT license.
+
+using System.Diagnostics;
+
+namespace System.Runtime.CompilerServices
+{
+    // Use this attribute alongside CompExactlyDependsOnAttribute to indicate that a method has
+    // a functional fallback path and should still be compiled into the Ready2Run image even when
+    // none of the CompExactlyDependsOn instruction sets are supported.
+    [AttributeUsage(AttributeTargets.Method | AttributeTargets.Constructor, AllowMultiple = false, Inherited = false)]
+#if MONO
+    [Conditional("unnecessary")] // Mono doesn't use Ready2Run so we can remove this attribute to reduce size
+#endif
+    internal sealed class CompHasFallbackAttribute : Attribute
+    {
+    }
+}
diff --git a/src/libraries/System.Private.CoreLib/src/System/SearchValues/SearchValues.cs b/src/libraries/System.Private.CoreLib/src/System/SearchValues/SearchValues.cs
index 73de2d970c626d..ecbbea5f8744fc 100644
--- a/src/libraries/System.Private.CoreLib/src/System/SearchValues/SearchValues.cs
+++ b/src/libraries/System.Private.CoreLib/src/System/SearchValues/SearchValues.cs
@@ -301,6 +301,7 @@ internal interface IRuntimeConst
         /// </summary>
         [MethodImpl(MethodImplOptions.AggressiveInlining)]
         [CompExactlyDependsOn(typeof(Ssse3))]
+        [CompHasFallback]
         internal static Vector128<byte> ShuffleNativeModified(Vector128<byte> vector, Vector128<byte> indices)
         {
             if (Ssse3.IsSupported)
diff --git a/src/libraries/System.Private.CoreLib/src/System/Text/Ascii.Utility.cs b/src/libraries/System.Private.CoreLib/src/System/Text/Ascii.Utility.cs
index cdec5bb675b312..6c84d46a1ab6e0 100644
--- a/src/libraries/System.Private.CoreLib/src/System/Text/Ascii.Utility.cs
+++ b/src/libraries/System.Private.CoreLib/src/System/Text/Ascii.Utility.cs
@@ -1632,6 +1632,7 @@ private static bool AllCharsInVectorAreAscii<T>(Vector128<T> vector)
 
         [MethodImpl(MethodImplOptions.AggressiveInlining)]
         [CompExactlyDependsOn(typeof(Avx))]
+        [CompHasFallback]
         private static bool AllCharsInVectorAreAscii<T>(Vector256<T> vector)
             where T : unmanaged
         {