Very very difficult.
I have a similar problem where a neighbours canes are running into my strawberry bed. But the strawberry bed is two years old so I will tolerate it for another year before forking the whole area. I'd be moving the bed anyway. Asparagus beds are probably best left for many years so that is not really an option.
Any physical attempt to fork out the Raspberries is likely to seriously damage your asparagus. But if you try a chemical attack, say by using say roundup gel on the emerging shoots you may well end up killing the canes on your neighbours plot. That is hardly likely to keep you on good terms with the next plot holder.
I'd suggest digging a narrow and very deep trench between your asparagus bed and your neighbours canes to sever all the roots running between the two. Once you have done this then get to work (carefully) with a glyphosate gel. I don't normally resort to chemicals like this, but I cannot see any other solution in this case.