If you’re into the history and tradition of Shibari (actually far less historical and traditional than many people think!), then you’ll want to use natural fibre rope because that’s what the samurai used...

There are better reasons. The right kinds of natural fibre rope are soft, flexible, grip well and form good tight knots. They feel good, and they’re a pleasure to work with. Above all, perhaps, they have a high “burn speed” over bare skin. If you pull synthetic rope across skin under any kind of tension, you will get “rope burns” very easily. That makes rope handling while tying (and even more so during the equally important untying) slow and difficult. Watching out for rope burns can destroy the natural rhythm of a rope scene. Well-prepared natural fibre rope is much better behaved in that respect, and that’s a large part of the reason why riggers who have got used to natural fibre rope are often reluctant to go back to using other kinds of rope.

Synthetic ropes have their place and I wouldn’t want to dispense with them altogether. But the more you use natural fibre rope, the less you want to use anything else. It just gets into your soul...