Natural stones, doesn't mean necessarily jnats. Even if you can get quite cheap jnat (if you get small ones or odd shapes), you can get even lesser expensive natural stones, maybe around you ? arkansas, coticule, but maybe some sandstone around you (i tried it and surely not amazing stones but it was fun) ?
They will be slower than most synthetic, they will be more expensive (jnats), they can be not consistent. But the feel for sharpening or polishing is really above synths. For polishing they really give a finish than no synthetic can give you, like a scratchless finish on the steel for exemple (there is scratches but they are blend by the effect of the natural stone). For sharpening... it's not as spectacular I would say, still the feel is better than synths and you can get some really great edges with good refinement and still some teeth to it.
Aizu are great but they are not that easy to get or cheap now. Less famous, natsuya (coarser than aizu generally), tajima, binsui yeah (coarser than aizu too) or some odd or small shape awasedo can be a way to try jnat. Coticule or belgium blue can also be a way if you're not specifically interested in jnats, they are really great naturals stones, especially for sharpening. Arkansas too but I know nothing about them.