I am very biased against any fabric of any kind in the garden so I am trying to disregard the fabric suggestion. Although, in this application it would make sense albeit ugly.
Have you talked with your neighbors? In any boundary problems it is imperative you strike up a conversation with your neighbors. You are not allowed to cut or trim in anyway that will harm any plant growing on their side of the fence, even weeds without incurring major ramifications. If your neighbors got huffy they would hold the upper hand.
From your point of view the problem is maintenance of this 'fence' as well as controlling the ivy and other plants from intruding on your property.
What I would do I'd dig a narrow trench down your side of the fence. Install some galvanized metal, 10" deep. That will help stop outside plants from growing into your yard. I doubt that you would be able to kill that ivy but do be careful if you run into critical root/trunk systems. I would purchase, if you do not already own, a great reciprocating hedger. I prefer gas powered but electric will do to keep the ivy a neat looking fence. Far better than that plastic.
It is a beautiful but higher than normal fence material. Use that ivy as a backdrop to some stunning perennials such as Crocosmia, the yellow one with a soft hedge of Quick Silver Hebe (need your zone) in front of your perennials. That is just one idea.
Keep your plants away from that fence by a foot so you have easy access to trim your 'fence' 3 or 4 times per season. I have been involved in the most insane and ludicrous law suits between neighbors dealing with less that you are now. The best thing you can do is to make lemonade from the lemons so to speak.
I would also go look up 'noxious weed management' for your area/country. In our country, English Ivy is one of the top ten worst weeds, seriously. You could use that in your communication with your neighbor as a bit of power, and use it judiciously. Nicely.
Also, thorny plants make excellent barriers for 'bad' guys. I don't see those plants in the picture but if they are there all the better. I hope this helps. Nothing is easy dealing with a boundary between neighbors. And a little bit more exercise is worth a good relationship with any neighbor as well as not being dragged into court and the ensuing ramifications! No one wins 'cept the lawyers.