The spun-bonded nature of Tyvek is a distinguishing characteristic that you are not going to find in a woven material, no matter how the woven material is other-wise treated. It is this spun-bonded quality that makes for excellent resistance to air infiltration, and using a woven material is most likely going to end up short in that department.
So let me challenge you on your no artificial materials requirement. Would it be alright to use an artificial material if it was on its way to clog up a landfill? Would it be good if you could reduce waste somewhere else by recycling and reusing? Because there are many, many mailing envelopes made of this same spun bonded polyethylene, and if you started collecting those and using sheets of them, it wouldn't be long before you had
enough area to wrap a tiny house. Polyethylene can be solvent welded with both acetone and toluene, so it might take a little experimentation to get the overlapping and gluing process down right, but you could make your own do-it-at-home Tyvek and be green in the process.