Damp proof membrane for shed wall

I wouldn't. You really need air flow both sides of the wood to avoid rot

I purchased a 14' x 8' shed a year last March. When I went to have a look at the product I notice that they lined the frame with Sisal Kraft Paper, brown paper with string and tar brown paper. I supplied them with a 50mtr x 1mtr roll of Weathershield breather underlay felt. They fitted this at no extra charge, all the way round and under the roof boards. I've also laid the same under the insulation to laminate flooring. The out side of the shiplap is painted with duck stain paint. I don't like the green mineral felt on the roof, I was considering taking it off and re roofing with cedar shingles. But cant be bothered at the moment. I'll wait till the mineral felt gives up.

I imagined a suitable product would be breathable to avoid this. Anyone?

Just use breathable roofing membrane. It is what it is for! Obviously plan it so that any water coming off drains to the outside.

Christian.