iOS meeting screen share: publish ReplayKit broadcast into the SFU room

Gap: on iOS, screen share only worked in native CallKit calls. In a scheduled /
code-joined SFU meeting the webview called setScreenShareEnabled() ->
getDisplayMedia(), which WKWebView does not implement, so it failed silently.
Production meetings are real LiveKit rooms, so we can publish natively instead.

Fix (native ReplayKit -> same LiveKit room as a dedicated screen participant):
- server: /api/meetings/token accepts screen:true -> mints a distinct
  <peerId>-screen identity (LiveKit allows one connection per identity), so the
  native publisher doesn't collide with the webview's own connection.
- native plugin: startMeetingScreenShare/stopMeetingScreenShare + connectScreenRoom
  (a screen-only LiveKit connection: mic off, no camera, no callConnected) that
  reuses the existing broadcast-extension publishing path.
- webview: toggleScreen routes to the native method on iOS SFU meetings; the
  screenShareState listener now drives the SFU case too (and tears the screen
  connection down on the system "Stop Broadcast"); sfuAttach/sfuDetach map the
  '<peerId>-screen' participant's screen track onto the sharer's tile and suppress
  the phantom person tile + the sharer's own self-view.

Web/server deploy now; the native method needs the next Codemagic build to test
on device. Verified: db-smoke 22/22; Swift braces balanced.

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

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

@@ -34,6 +34,8 @@ public class NativeCallPlugin: CAPPlugin, CAPBridgedPlugin, PKPushRegistryDelega
         CAPPluginMethod(name: "switchCamera", returnType: CAPPluginReturnPromise),
         CAPPluginMethod(name: "startScreenShare", returnType: CAPPluginReturnPromise),
         CAPPluginMethod(name: "stopScreenShare", returnType: CAPPluginReturnPromise),
+        CAPPluginMethod(name: "startMeetingScreenShare", returnType: CAPPluginReturnPromise),
+        CAPPluginMethod(name: "stopMeetingScreenShare", returnType: CAPPluginReturnPromise),
         CAPPluginMethod(name: "setHolePunch", returnType: CAPPluginReturnPromise),
         CAPPluginMethod(name: "setTileZoom", returnType: CAPPluginReturnPromise),
         CAPPluginMethod(name: "syncVideoTiles", returnType: CAPPluginReturnPromise),
@@ -391,6 +393,41 @@ public class NativeCallPlugin: CAPPlugin, CAPBridgedPlugin, PKPushRegistryDelega
         call.resolve()
     }
 
+    // Screen share into a SCHEDULED / code-joined SFU meeting. The WebView already holds the meeting
+    // connection (identity = peerId) and WKWebView has no getDisplayMedia — so we open a SECOND, screen-ONLY
+    // native LiveKit connection under a distinct `<peerId>-screen` identity and publish the ReplayKit
+    // broadcast into the SAME room. The WebView maps that identity back onto the sharer's tile.
+    @objc func startMeetingScreenShare(_ call: CAPPluginCall) {
+        let url = call.getString("url") ?? ""
+        let token = call.getString("token") ?? ""
+        guard !url.isEmpty, !token.isEmpty else { call.reject("url/token required"); return }
+        connectScreenRoom(url: url, token: token)
+        DispatchQueue.main.async { BroadcastManager.shared.requestActivation() } // system broadcast picker
+        call.resolve()
+    }
+
+    @objc func stopMeetingScreenShare(_ call: CAPPluginCall) {
+        BroadcastManager.shared.requestStop()
+        let r = room; room = nil
+        Task { await r?.disconnect() } // drop the screen-only connection (no call/tile/transcriber side effects)
+        call.resolve()
+    }
+
+    // A screen-ONLY LiveKit connection (mic off, no camera) used purely to publish the ReplayKit broadcast
+    // into an SFU meeting. Unlike connectRoom() it enables no mic and fires no callConnected — this is not a call.
+    private func connectScreenRoom(url: String, token: String) {
+        guard !url.isEmpty, !token.isEmpty else { return }
+        let old = room
+        let opts = RoomOptions(defaultScreenShareCaptureOptions: ScreenShareCaptureOptions(useBroadcastExtension: true))
+        let r = Room(roomOptions: opts)
+        room = r
+        Task { [weak self] in
+            await old?.disconnect() // never leave a duplicate connection
+            do { try await r.connect(url: url, token: token) }
+            catch { self?.notifyListeners("screenShareState", data: ["sharing": false, "error": String(describing: error)]) }
+        }
+    }
+
     // MARK: - #5 Live transcript (native SFSpeechRecognizer)
 
     // The current published local mic track — the source we tap for transcription. nil until the user has