Isn't it basic inter-process communication? It's the same as with a chat client/server. You have a server to which the client has a network connection through a socket. (there are variants for this, even websockets, you can find more info about this through google and check what you need, e.g. named pipes might be easiest if you are on the same network). Your phone application calls a service, the service does the work and calls the server the client has connection to to send the clients a message that things have been updated. The client receives over its network connection the message from the server and refreshes.
There are various ways to perform interprocess communication, and you need 2-way communication, not call-respond, so essentially the same as a basic chat-client/server, which are in its simplest form build on sockets, but there are other ways.