What can I do to keep high hydration dough from sticking to my hands?

Let the dough rest in the bowl for a while after mixing. Time will not only improve the flavor (insert autolyse proselytism here) but will also give the flour time to hydrate, making the dough somewhat easier to handle.

Dough will become less sticky as you work it -- witness the way dough kneaded in a mixer or food processor initially sticks to the sides but eventually forms a ball and leaves the bowl clean. So, one option is to work it a bit with a mixer first, or just with a wooden spoon in the bowl.

Anil's suggestion to oil your hands is a good one. If the kind of dough you're making allows it, add some oil or butter to the dough during mixing, too.

Flour helps, of course. Instead of trying to coat your hands in flour, throw a bit on the kneading surface and on the top of the dough.

For very sticky dough, a dough knife can help you scrape the dough off the surface and fold it over onto itself until it becomes easier to handle. This is easiest if you're working on a hard, smooth kneading surface like marble.

Oil was the answer for me when trying to shape my rye bread. As the rye flour is stickier then bread flour which in turn makes the blended flours stickier then normal. It made the dough as workable as my regular dough from bread flour alone.
My experience is in using the no knead method. Using the oil on my hand actually worked very well. I had zero dough on my hands between the first and second proofing.

There are two main ways this is accomplished, and one condition which will cause stickiness.

To mitigate sticking:

1. Use flour
2. Use water

I usually knead by hand, and keeping my hands generously moist is often enough to prevent sticking (I knead in a bowl). This causes the dough to be a little wetter than I aim for.

Then, while shaping, I will use flour on the surface, since I won't be knead it anymore.

I will put some oil on the baking parchment when baking, to prevent sticking.

What causes dough to stick overly much is not sufficient hydration (time) and gluten formation (time & kneading)

My suggestion is to try it with grease or oil, similar to oiling utensils. This is what you can do when you prepare dough. If you are concerned about the amount of oil in the bread, you can try flour.

Apart from this, the dough consistency plays a major role. If you make your dough a little harder, the stickiness will decrease. If it is too loose, it will stick a lot.

An additional tip: kneading faster keeps the dough from sticking too badly. For some reason, speed makes a noticeable difference.

Also, just go with it. At the end, put a little flour on your hands and rub them together; the little dough stuck to your hands is dried out by the extra flour and crumbles off your hands into your dough, so you don't lose any dough.

Embrace the sticking. You can do a stretch-and-fold entirely in your hands, and instead of gripping the dough, you let the two ends stick to your hands while separating them and closing them again. It aligns the gluten beautifully. When you are ready, you have to separate/scrape off the dough from your hands. If you need a round boule at this time, you may need to do it in a basin of flour, but for a ciabatta, it is enough to do the triple fold.