diff --git a/Library/Homebrew/cask/caskroom.rb b/Library/Homebrew/cask/caskroom.rb
index bef02c7461c12..5aac963b10eb8 100644
--- a/Library/Homebrew/cask/caskroom.rb
+++ b/Library/Homebrew/cask/caskroom.rb
@@ -22,6 +22,12 @@ def self.paths
     end
     private_class_method :paths
 
+    # Return all tokens for installed casks.
+    sig { returns(T::Array[String]) }
+    def self.tokens
+      paths.map(&:basename).map(&:to_s)
+    end
+
     sig { returns(T::Boolean) }
     def self.any_casks_installed?
       paths.any?
@@ -46,18 +52,14 @@ def self.ensure_caskroom_exists
 
     sig { params(config: T.nilable(Config)).returns(T::Array[Cask]) }
     def self.casks(config: nil)
-      paths.sort.map do |path|
-        token = path.basename.to_s
-
-        begin
-          CaskLoader.load(token, config: config)
-        rescue TapCaskAmbiguityError
-          tap_path = CaskLoader.tap_paths(token).first
-          CaskLoader::FromTapPathLoader.new(tap_path).load(config: config)
-        rescue CaskUnavailableError
-          # Don't blow up because of a single unavailable cask.
-          nil
-        end
+      tokens.sort.map do |token|
+        CaskLoader.load(token, config: config)
+      rescue TapCaskAmbiguityError
+        tap_path = CaskLoader.tap_paths(token).first
+        CaskLoader::FromTapPathLoader.new(tap_path).load(config: config)
+      rescue
+        # Don't blow up because of a single unavailable cask.
+        nil
       end.compact
     end
   end
diff --git a/Library/Homebrew/cmd/untap.rb b/Library/Homebrew/cmd/untap.rb
index bf12b54b8dca1..99af3c2638c39 100644
--- a/Library/Homebrew/cmd/untap.rb
+++ b/Library/Homebrew/cmd/untap.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "cli/parser"
@@ -19,15 +19,45 @@ def untap_args
     end
   end
 
+  sig { void }
   def untap
     args = untap_args.parse
 
     args.named.to_installed_taps.each do |tap|
       odie "Untapping #{tap} is not allowed" if tap.core_tap? && Homebrew::EnvConfig.no_install_from_api?
 
-      if Homebrew::EnvConfig.no_install_from_api? || (!tap.core_tap? && tap != "homebrew/cask")
-        installed_tap_formulae = Formula.installed.select { |formula| formula.tap == tap }
-        installed_tap_casks = Cask::Caskroom.casks.select { |cask| cask.tap == tap }
+      if Homebrew::EnvConfig.no_install_from_api? || (!tap.core_tap? && !tap.core_cask_tap?)
+        installed_formula_names = T.let(nil, T.nilable(T::Set[String]))
+        installed_tap_formulae = tap.formula_names.map do |formula_name|
+          # initialise lazily in case there's no formulae in this tap
+          installed_formula_names ||= Set.new(Formula.installed_formula_names)
+          next unless installed_formula_names.include?(formula_name)
+
+          formula = begin
+            Formulary.factory("#{tap.name}/#{formula_name}")
+          rescue
+            # Don't blow up because of a single unavailable formula.
+            next
+          end
+
+          formula if formula.any_version_installed?
+        end.compact
+
+        installed_cask_tokens = T.let(nil, T.nilable(T::Set[String]))
+        installed_tap_casks = tap.cask_tokens.map do |cask_token|
+          # initialise lazily in case there's no casks in this tap
+          installed_cask_tokens ||= Set.new(Cask::Caskroom.tokens)
+          next unless installed_cask_tokens.include?(cask_token)
+
+          cask = begin
+            Cask::CaskLoader.load("#{tap.name}/#{cask_token}")
+          rescue
+            # Don't blow up because of a single unavailable cask.
+            next
+          end
+
+          cask if cask.installed?
+        end.compact
 
         if installed_tap_formulae.present? || installed_tap_casks.present?
           installed_names = (installed_tap_formulae + installed_tap_casks.map(&:token)).join("\n")
diff --git a/Library/Homebrew/formula.rb b/Library/Homebrew/formula.rb
index ddd868f888b8b..8ac85affdd6c1 100644
--- a/Library/Homebrew/formula.rb
+++ b/Library/Homebrew/formula.rb
@@ -1950,6 +1950,13 @@ def self.racks
     end
   end
 
+  # An array of all currently installed formula names.
+  # @private
+  sig { returns(T::Array[String]) }
+  def self.installed_formula_names
+    racks.map(&:basename).map(&:to_s)
+  end
+
   # An array of all installed {Formula}
   # @private
   def self.installed