The best open source alternative to Microsoft Remote Desktop is DWService, which is both free and Open Source. If that doesn't suit you, our users have ranked more than 25 alternatives to Microsoft Remote Desktop and five of them is open source so hopefully you can find a suitable replacement. Other interesting open source alternatives to Microsoft Remote Desktop are Dayon! (Free, Open Source), WifiMouse (Free, Open Source), screensy (Free, Open Source) and MyRDP (Free, Open Source).
DWService is an open source project which offers a service that allows access to remote systems (Windows, Mac, Linux, Raspberry...) using a standard web browser - without any component required on the client side.
Dayon! is a user friendly, open-source, cross-platform solution, that allows controlling of remote computers. It can be used as free alternative to various commercial remote desktop and remote assistance products.