I've finally come across a material that works as a wrapping to make a slippery surface sticky. It's used for this by other people, and after a bit of testing, it seems to work. (I haven't yet used it enough to know whether it slowly slips like silicone rubber does.)
The material I have in hand is Shushi double sided fabric backed adhesive tape (probably not widely available, I think it's the company's tape 99 in China). It feels like tarred fabric and is slightly stretchy. It is not very sticky but has a bit of grip when wrapped tightly. I gather double sided hockey tape is the same material and would also work, so long as the adhesive can grip low surface energy plastics.