I have an active splitter for my DAB/FM radio. It takes the aerial plug from the MMC factory one you describe and provides flying leads to FM and DAB sockets. Its powered by 12v from the electric aerial feed on radio's loom so is only active when radio is on. Inexpensive one here.

Richard

2018 Roadster 3.7
1966 Land Rover S2a 88
2024 Royal Enfield Guerrilla 450
1945 Guzzi Airone