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When I did the shed, a guide I found suggested torch-on roofing felt - the sort you'd use on a flat roof. Logic being its much, much tougher than the stuff "designed for" sheds.
I may chuck a load on the top as soon as I finish the roof - that way, I don't need to complete the live roof straight away, and have more time for dithering.
NotThamesWater-
Here's the guide I used:
https://www.fixmyroof.co.uk/how-to-felt-a-shed-roof/#
You can get different colours. I wanted green, but black/grey was cheapest*. Mine was seconds mineral torch-on stuff from ebay/some guys garden. One end was a bit knackered from being stood up on its end and another section was creased, but I just positioned it so those bits got trimmed off. Collecting was a bit more faff but it still cost the same/less than the thin shed stuff from Wickes.
But if I were you I'd throw on some cut up blue Ikea bags as an incentive to complete the living roof....
... although you could always put the living roof on top (pending weight restrictions).
*and matched the other sheds, preventing mission creep and wrath from Mrs. H. who already thinks 3 sheds and a brick outhouse is too much
hugo7
When I did the shed, a guide I found suggested torch-on roofing felt - the sort you'd use on a flat roof. Logic being its much, much tougher than the stuff "designed for" sheds.
EBay has loads of cheap stuff either left over from roof jobs, or isn't up to par for a resi flat roof but fine for a shed.
Living roof is infinitely better, but thought I'd mention it for others.