I have just been looking through the VAG system information, the original page I had for the Jetta showed the DAB aerial is in the shark fin this was confusing as to get to the right page I needed to navigate to Golf Saloon. This fits with your description and the shark fin is used for navigation and telephone systems, I now have I believe the system on your vehicle.
This shows a left aerial and a right aerial coming from the radio. Both aerials have an impedance converter in the circuit and then go to the rear screen aerial, this is earthed via a Frequency Modulation filter on both sides. This I think would be the first check to make, as if these blocks were not earthing correctly it would give the intermittent reception you have.
All the components are in the rear area below the window, the full list is:
Frequency modulation filter (negative wire), Amplitude modulation filter, frequency modulation filter, right and left aerial module (these go to earth and are screwed to the body) the aerial itself which is in the rear window.
I hope this helps but do get back to me again if needed, and do let me know how you get on.