Outgoing iOS screen-share via ReplayKit broadcast extension

Lets a native-call user share their iPhone screen (whole device, works
backgrounded). LiveKit 2.15.3 ships the broadcast stack (BroadcastManager +
LKSampleHandler + IPC), so:

- New Broadcast Upload Extension target "BroadcastExtension"
  (com.bizgaze.connect.broadcast): SampleHandler.swift subclasses
  LKSampleHandler; injected by mobile/scripts/add-broadcast-extension.rb
  which also LINKS the LiveKit SPM product into the extension + sets the
  App Group. Sources in mobile/ios-broadcast/.
- Plugin: Room created with ScreenShareCaptureOptions(useBroadcastExtension:
  true); startScreenShare -> BroadcastManager.requestActivation() (system
  picker); stopScreenShare -> requestStop(); BroadcastManagerDelegate ->
  fires screenShareState to the web. LiveKit auto-publishes the track.
- ios-patch.sh: RTCScreenSharingExtension + RTCAppGroupIdentifier keys.
- codemagic.yaml: run the injector + sign the 3rd bundle id (.broadcast).
- home.html: toggleScreen native -> start/stop; screenShareState listener
  reflects state + broadcasts meeting-screen so peers' stage shows it.

REQUIRES a one-time manual Apple portal step: enable the App Group on the
com.bizgaze.connect.broadcast App ID (see mobile/IOS_SETUP.md) or the
archive fails code-signing. SPM-linked extension is new on our CI — expect
build iteration. Web deployed (no-ops on builds without startScreenShare).

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

mobile/plugins/native-call/ios/Sources/NativeCallPlugin/NativeCallPlugin.swift

@@ -19,7 +19,7 @@ import LiveKit // SPM product name is 'LiveKit' (Package.swift). (The old CocoaP
 //     the token from the WebView via reportOutgoingCall(). The WebView is UI only for native calls (it must
 //     NOT also join the room — LiveKit allows one connection per identity).
 @objc(NativeCallPlugin)
-public class NativeCallPlugin: CAPPlugin, CAPBridgedPlugin, PKPushRegistryDelegate, CXProviderDelegate {
+public class NativeCallPlugin: CAPPlugin, CAPBridgedPlugin, PKPushRegistryDelegate, CXProviderDelegate, BroadcastManagerDelegate {
     public let identifier = "NativeCallPlugin"
     public let jsName = "NativeCall"
     public let pluginMethods: [CAPPluginMethod] = [
@@ -29,6 +29,8 @@ public class NativeCallPlugin: CAPPlugin, CAPBridgedPlugin, PKPushRegistryDelega
         CAPPluginMethod(name: "setMuted", returnType: CAPPluginReturnPromise),
         CAPPluginMethod(name: "setCamera", returnType: CAPPluginReturnPromise),
         CAPPluginMethod(name: "switchCamera", returnType: CAPPluginReturnPromise),
+        CAPPluginMethod(name: "startScreenShare", returnType: CAPPluginReturnPromise),
+        CAPPluginMethod(name: "stopScreenShare", returnType: CAPPluginReturnPromise),
         CAPPluginMethod(name: "syncVideoTiles", returnType: CAPPluginReturnPromise),
         CAPPluginMethod(name: "endCall", returnType: CAPPluginReturnPromise)
     ]
@@ -75,6 +77,10 @@ public class NativeCallPlugin: CAPPlugin, CAPBridgedPlugin, PKPushRegistryDelega
         // preferSpeaker and fixed it). Listening for route changes makes that self-healing.
         NotificationCenter.default.addObserver(self, selector: #selector(audioRouteChanged(_:)),
                                                name: AVAudioSession.routeChangeNotification, object: nil)
+
+        // Screen sharing (ReplayKit broadcast extension). LiveKit tells us when a broadcast starts/stops and
+        // (with shouldPublishTrack=true, the default) auto-publishes/unpublishes the screen-share track.
+        BroadcastManager.shared.delegate = self
     }
 
     @objc private func audioRouteChanged(_ note: Notification) {
@@ -91,7 +97,10 @@ public class NativeCallPlugin: CAPPlugin, CAPBridgedPlugin, PKPushRegistryDelega
         // engine in didActivate can block on first use — determining it up front avoids that.
         AVAudioSession.sharedInstance().requestRecordPermission { _ in }
         let old = room
-        let r = Room()
+        // Route screen-share through the ReplayKit broadcast extension (so the user can share their screen
+        // even when the app is backgrounded, and it captures the whole phone, not just the WebView).
+        let opts = RoomOptions(defaultScreenShareCaptureOptions: ScreenShareCaptureOptions(useBroadcastExtension: true))
+        let r = Room(roomOptions: opts)
         room = r
         Task { [weak self] in
             await old?.disconnect() // drop any previous/stale connection so we never leave DUPLICATE participants
@@ -331,6 +340,26 @@ public class NativeCallPlugin: CAPPlugin, CAPBridgedPlugin, PKPushRegistryDelega
         }
     }
 
+    // Screen sharing from iOS: show the system broadcast picker. When the user starts the broadcast, the
+    // extension streams the screen to us over IPC and LiveKit publishes it (BroadcastManager.shouldPublishTrack
+    // defaults true). broadcastManager(didChangeState:) fires screenShareState back to the web either way.
+    @objc func startScreenShare(_ call: CAPPluginCall) {
+        guard room != nil else { call.reject("no active call"); return }
+        DispatchQueue.main.async { BroadcastManager.shared.requestActivation() } // presents RPSystemBroadcastPickerView
+        call.resolve()
+    }
+
+    @objc func stopScreenShare(_ call: CAPPluginCall) {
+        BroadcastManager.shared.requestStop()
+        call.resolve()
+    }
+
+    // MARK: - BroadcastManagerDelegate
+
+    public func broadcastManager(didChangeState isBroadcasting: Bool) {
+        notifyListeners("screenShareState", data: ["sharing": isBroadcasting])
+    }
+
     // Position native video views to match the web meeting tiles. `tiles` = [{uid, local, x, y, w, h}] in
     // CSS px (== points; getBoundingClientRect coords). We create/move a VideoView for each participant that
     // has a live camera track, and remove views for tiles that are gone or whose camera is off (so the web