diff --git a/Library/Homebrew/cmd/tap.rb b/Library/Homebrew/cmd/tap.rb
index f3f751ead323b..88953928fb15a 100644
--- a/Library/Homebrew/cmd/tap.rb
+++ b/Library/Homebrew/cmd/tap.rb
@@ -60,7 +60,7 @@ def tap
         tap.fix_remote_configuration
       end
     elsif args.no_named?
-      puts Tap.select(&:installed?)
+      puts Tap.select(&:installed?).sort_by(&:name)
     else
       tap = Tap.fetch(args.named.first)
       begin
diff --git a/Library/Homebrew/tap.rb b/Library/Homebrew/tap.rb
index c22a1325a4cd9..06f4dde26e3cf 100644
--- a/Library/Homebrew/tap.rb
+++ b/Library/Homebrew/tap.rb
@@ -5,6 +5,7 @@
 require "completions"
 require "extend/cachable"
 require "description_cache_store"
+require "set"
 require "settings"
 
 # A {Tap} is used to extend the formulae provided by Homebrew core.
@@ -401,6 +402,7 @@ def install(quiet: false, clone_target: nil, force_auto_update: nil,
 
     Commands.rebuild_commands_completion_list
     link_completions_and_manpages
+    Tap.registry << self
 
     formatted_contents = contents.presence&.to_sentence&.dup&.prepend(" ")
     $stderr.puts "Tapped#{formatted_contents} (#{path.abv})." unless quiet
@@ -511,6 +513,7 @@ def uninstall(manual: false)
 
     Commands.rebuild_commands_completion_list
     clear_cache
+    Tap.registry.delete(self)
 
     return if !manual || !official?
 
@@ -880,27 +883,32 @@ def ==(other)
     other = Tap.fetch(other) if other.is_a?(String)
     self.class == other.class && name == other.name
   end
+  alias eql? ==
 
-  def self.each(&block)
-    return to_enum unless block
+  sig { returns(Integer) }
+  def hash
+    [self.class, name].hash
+  end
 
-    installed_taps = if TAP_DIRECTORY.directory?
-      TAP_DIRECTORY.subdirs
-                   .flat_map(&:subdirs)
-                   .map(&method(:from_path))
-    else
-      []
-    end
+  # A set with all installed and core taps.
+  #
+  # @private
+  sig { returns(T::Set[Tap]) }
+  def self.registry
+    cache[:registry] ||= Set.new.tap do |set|
+      if TAP_DIRECTORY.directory?
+        set.merge TAP_DIRECTORY.subdirs
+                               .flat_map(&:subdirs)
+                               .map(&method(:from_path))
+      end
 
-    available_taps = if Homebrew::EnvConfig.no_install_from_api?
-      installed_taps
-    else
-      default_taps = T.let([CoreTap.instance], T::Array[Tap])
-      default_taps << CoreCaskTap.instance if OS.mac? # rubocop:disable Homebrew/MoveToExtendOS
-      installed_taps + default_taps
-    end.sort_by(&:name).uniq
+      set << CoreTap.instance
+      set << CoreCaskTap.instance if OS.mac? # rubocop:disable Homebrew/MoveToExtendOS
+    end
+  end
 
-    available_taps.each(&block)
+  def self.each(&block)
+    registry.each(&block)
   end
 
   # An array of all installed {Tap} names.
diff --git a/Library/Homebrew/test/tap_spec.rb b/Library/Homebrew/test/tap_spec.rb
index cc7b7f5edaa1f..0b48c3c6abc8c 100644
--- a/Library/Homebrew/test/tap_spec.rb
+++ b/Library/Homebrew/test/tap_spec.rb
@@ -140,10 +140,6 @@ def setup_completion(link:)
     end
   end
 
-  specify "::names" do
-    expect(described_class.names.sort).to eq(["homebrew/core", "homebrew/foo"])
-  end
-
   specify "attributes" do
     expect(homebrew_foo_tap.user).to eq("Homebrew")
     expect(homebrew_foo_tap.repo).to eq("foo")
@@ -404,10 +400,12 @@ def setup_completion(link:)
     setup_completion link: true
 
     tap = described_class.new("Homebrew", "bar")
+    expect(described_class.registry).not_to include(tap)
 
     tap.install clone_target: homebrew_foo_tap.path/".git"
 
     expect(tap).to be_installed
+    expect(described_class.registry).to include(tap)
     expect(HOMEBREW_PREFIX/"share/man/man1/brew-tap-cmd.1").to be_a_file
     expect(HOMEBREW_PREFIX/"etc/bash_completion.d/brew-tap-cmd").to be_a_file
     expect(HOMEBREW_PREFIX/"share/zsh/site-functions/_brew-tap-cmd").to be_a_file
@@ -415,6 +413,7 @@ def setup_completion(link:)
     tap.uninstall
 
     expect(tap).not_to be_installed
+    expect(described_class.registry).not_to include(tap)
     expect(HOMEBREW_PREFIX/"share/man/man1/brew-tap-cmd.1").not_to exist
     expect(HOMEBREW_PREFIX/"share/man/man1").not_to exist
     expect(HOMEBREW_PREFIX/"etc/bash_completion.d/brew-tap-cmd").not_to exist
@@ -501,6 +500,24 @@ def setup_completion(link:)
     end
   end
 
+  describe ".registry" do
+    it "includes the core and cask taps by default", :needs_macos do
+      expect(described_class.registry).to include(CoreTap.instance, CoreCaskTap.instance)
+    end
+
+    it "includes the core tap and excludes the cask tap by default", :needs_linux do
+      expect(described_class.registry).to include(CoreTap.instance)
+      expect(described_class.registry).not_to include(CoreCaskTap.instance)
+    end
+
+    it "ignores a second instance of the same tap" do
+      expect { described_class.registry << described_class.new("example", "first") }
+        .to change { described_class.registry.size }.by(1)
+      expect { described_class.registry << described_class.new("example", "first") }
+        .not_to(change { described_class.registry.size })
+    end
+  end
+
   describe "Formula Lists" do
     describe "#formula_renames" do
       it "returns the formula_renames hash" do