diff --git a/Library/Homebrew/os/mac/keg.rb b/Library/Homebrew/os/mac/keg.rb
index 30c76420b3..ebe0f17d7f 100644
--- a/Library/Homebrew/os/mac/keg.rb
+++ b/Library/Homebrew/os/mac/keg.rb
@@ -6,7 +6,7 @@ class Keg
   def initialize(path)
     super
 
-    @require_relocation = T.let(nil, T.nilable(T::Boolean))
+    @require_relocation = T.let(false, T::Boolean)
   end
 
   sig { params(id: String, file: Pathname).returns(T::Boolean) }
@@ -15,7 +15,7 @@ class Keg
 
     @require_relocation = true
     odebug "Changing dylib ID of #{file}\n  from #{file.dylib_id}\n    to #{id}"
-    file.change_dylib_id(id)
+    file.change_dylib_id(id, strict: false)
     true
   rescue MachO::MachOError
     onoe <<~EOS
@@ -32,7 +32,7 @@ class Keg
 
     @require_relocation = true
     odebug "Changing install name in #{file}\n  from #{old}\n    to #{new}"
-    file.change_install_name(old, new)
+    file.change_install_name(old, new, strict: false)
     true
   rescue MachO::MachOError
     onoe <<~EOS
@@ -49,7 +49,7 @@ class Keg
 
     @require_relocation = true
     odebug "Changing rpath in #{file}\n  from #{old}\n    to #{new}"
-    file.change_rpath(old, new)
+    file.change_rpath(old, new, strict: false)
     true
   rescue MachO::MachOError
     onoe <<~EOS
@@ -63,7 +63,7 @@ class Keg
   sig { params(rpath: String, file: MachOShim).returns(T::Boolean) }
   def delete_rpath(rpath, file)
     odebug "Deleting rpath #{rpath} in #{file}"
-    file.delete_rpath(rpath)
+    file.delete_rpath(rpath, strict: false)
     true
   rescue MachO::MachOError
     onoe <<~EOS
diff --git a/Library/Homebrew/os/mac/mach.rb b/Library/Homebrew/os/mac/mach.rb
index b44a13ab98..5a0a27566a 100644
--- a/Library/Homebrew/os/mac/mach.rb
+++ b/Library/Homebrew/os/mac/mach.rb
@@ -32,8 +32,9 @@ module MachOShim
       machos = []
       mach_data = []
 
-      if macho.is_a?(MachO::FatFile)
-        machos = T.cast(macho, MachO::FatFile).machos
+      case (macho = self.macho)
+      when MachO::FatFile
+        machos = macho.machos
       else
         machos << macho
       end
@@ -70,8 +71,8 @@ module MachOShim
 
   # TODO: See if the `#write!` call can be delayed until
   #       we know we're not making any changes to the rpaths.
-  sig { params(rpath: String).void }
-  def delete_rpath(rpath)
+  sig { params(rpath: String, strict: T::Boolean).void }
+  def delete_rpath(rpath, strict: true)
     candidates = rpaths(resolve_variable_references: false).select do |r|
       resolve_variable_name(r) == resolve_variable_name(rpath)
     end
@@ -79,25 +80,25 @@ module MachOShim
     # Delete the last instance to avoid changing the order in which rpaths are searched.
     rpath_to_delete = candidates.last
 
-    macho.delete_rpath(rpath_to_delete, { last: true })
+    macho.delete_rpath(rpath_to_delete, { last: true, strict: })
     macho.write!
   end
 
-  sig { params(old: String, new: String, uniq: T::Boolean, last: T::Boolean).void }
-  def change_rpath(old, new, uniq: false, last: false)
-    macho.change_rpath(old, new, { uniq: uniq, last: last })
+  sig { params(old: String, new: String, uniq: T::Boolean, last: T::Boolean, strict: T::Boolean).void }
+  def change_rpath(old, new, uniq: false, last: false, strict: true)
+    macho.change_rpath(old, new, { uniq:, last:, strict: })
     macho.write!
   end
 
-  sig { params(id: String).void }
-  def change_dylib_id(id)
-    macho.change_dylib_id(id)
+  sig { params(id: String, strict: T::Boolean).void }
+  def change_dylib_id(id, strict: true)
+    macho.change_dylib_id(id, { strict: })
     macho.write!
   end
 
-  sig { params(old: String, new: String).void }
-  def change_install_name(old, new)
-    macho.change_install_name(old, new)
+  sig { params(old: String, new: String, strict: T::Boolean).void }
+  def change_install_name(old, new, strict: true)
+    macho.change_install_name(old, new, { strict: })
     macho.write!
   end
 
diff --git a/Library/Homebrew/os/mac/sdk.rb b/Library/Homebrew/os/mac/sdk.rb
index e5ea0f2a19..498fd8e3bb 100644
--- a/Library/Homebrew/os/mac/sdk.rb
+++ b/Library/Homebrew/os/mac/sdk.rb
@@ -38,7 +38,7 @@ module OS
 
       sig { void }
       def initialize
-        @all_sdks = T.let([], T::Array[SDK])
+        @all_sdks = T.let(nil, T.nilable(T::Array[SDK]))
         @sdk_prefix = T.let(nil, T.nilable(String))
       end
 
@@ -52,7 +52,9 @@ module OS
 
       sig { returns(T::Array[SDK]) }
       def all_sdks
-        return @all_sdks unless @all_sdks.empty?
+        return @all_sdks if @all_sdks
+
+        @all_sdks = []
 
         # Bail out if there is no SDK prefix at all
         return @all_sdks unless File.directory? sdk_prefix