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Thanks. Well, sand or wire brush off the remaining loose render/bits of sand and cement, rather than the brickwork itself. It's very porous so I'm worried about too much moisture getting in if I just leave it.
It's not just the cost of getting it rendered, I would of course need the neighbour's permission to do her side. And I reckon she'd say no even if I offered to pay for it all, she's not the type to get anything done.
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What's your neighbour going to do if you just cover her side too.
Just rendering it to the halfway point is going to create a nice water retaining barrier that will prevent drying out of the brick on your side, with a nice water entry point on your side.
Is it a solid or a cavity wall, and where is the damp proof course on it?
Sand the brickwork down? It probably needs to be stripped back and re-rendered. If you’re more worried about the neighbour not contributing than you are about fixing it, just leave it as it is.