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Unless you are going to go a proper rally tyre (no idea about these, but I imagine expensive and fast wearing on anything that isn't dirt) your only other option for an off road tyre which typically come wide with a very tall aspect ratio (eg 265/75 R17) which is more than likely not going to fit under your guards ;)

Having taken both my Skyline (aftermarket 17" wheels) and Liberty (factory 18" wheels) bush on many occasions I don't see why one couldn't use a 17" with a "all round" style tread. If it really bothered me I'd consider a smaller wheel to allow a higher profile tyre to absorb whatever bumps there are on the road...

If you want to go fast on dirt... I think you should take a quick look at the kind of cars that enter WRC ;)

i'd go back to the smallest diameter wheels with regular tyres.

x2 for the smallest lightest tough wheels and tyres you can find. So 14 x 6 with 185/60R14 tyres if it's an NA. I'm doing the occasional bit of dirt in mine.

Going to much stiffer springs on my Teins (up from 5/4 to 10/7 Kgmm, where stock is around 3/1.5) has made it surprisingly better planted on dirt and corrugations.