Thanks to Richard's help ryy the issue is resolved now.
I followed this guide, had to change the authentication pattern in the Virtual Proxies tab from "Forms" (which i specified some time ago) to default "Windows". Restarted the reload tasks. Profit!
The guide above says that at least one virtual proxy has to contain "Windows" authentication pattern.
I've written up the instructions to create a second virtual proxy to allow the REST connectors to use this whilst allowing users to access with Forms authentication.