I would use RabbitMQ and consume events forwarding them as push notifications. This will work while the user is not actively connected to the website and enhance the engagement with each user experience that will return to the website when notified for more information see How to setup basic web push notification functionality using a Flask backend or How to send push notifications to a browser in ASP.NET Core or Sending Notifications with Spring Boot, Angular, and Firebase Cloud Messaging this way the RabbitMQ will not wait until the user is back online. If the user is online you can forward the notification directly to the Xamarin application via WebSockets and a load balancer like NGINX that can handle many WebSockets in an optimized way.
Synchronous API polling from the client-side is the less preferred way since it overloads the webserver with requests while nothing was changed.