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Note it is not included in the title plan of either property
Half the passage is owned by each house, the red line always falls inside the boundary.
There will be another drawing showing points C and D and the brown bit that will make it all clear. Looks like there’s a side passage plus a rear lane so it’s a bit confusing without the other plan.
ETA: all normal, sensible arrangements - someone owns the land, others have rights to cross it.
IANAL but it sounds like (i) you have right of way over the strip, (ii) neighbour has obligation to maintain it and (iii) unknown third party (“Vendor”) actually owns it?
Note it is not included in the title plan of either property.
The lawyer you are paying to handle this stuff should be able to give you a better answer, backed by a PI insurance policy.