Here's an 8-year-old article by Ryan Jordan on tarp techniques in bad weather.
I don't have extensive tarp experience in bad weather, but I did use my (now-discontinued) BPL Nano tarp for 3 rainy nights on the Tahoe Rim Trail last year. One of the nights was just a light drizzle, while the other two were 35mph winds with rain and 5 minutes periods of snow and hail. One of those nights was spent mostly sheltered by large trees, but the other was pretty exposed near a lake at 8000'.
I had a bivy, which helped, as there was a bit of splash at the higher pitched head end of the tarp. I got into camp just before dark and campsite selection was limited, so I wasn't able to properly pitch the tarp foot-end into the wind. It was more of a crosswind. A beak would have served the same purpose for splash protection, I think, but I'd take a bivy over a beak, as the bivy adds a bit of warmth to my sleep system whereas the beak would not. But in warmer, more humid weather, the bivy might lead to claminess and/or condensation in your sleeping bag or quilt.
Beyond the bit of splash at the head, I was completely safe and secure as hail crashed down, and it was kind of cool to be so exposed yet protected from such wild weather. One of the advantages of cuben is that I did not have to get up to retension my lines as the night went on, since cuben does not stretch.
To me, the biggest drawback of a tarp in that kind of weather is that it has to be pitched low, (see the pictures of Dr. Jordan in the linked article)leaving little room for manuevering. Not a problem if all you are going to do is sleep. A bit of a pain though getting ready in the morning, and not much room for eating or cooking w/o sitting out in th rain.
If starting again, I might go with something like the MLD Trailstar(solo) + Innernet combo in cuben for maximum protection + room. Or one of the Zpacks tarps.
I thought about the slightly more limited pitch options with a cat-cut tarp vs a flat tarp, but then thought, 'How many pitch options would I ever really use beyond an A-Frame in it's various forms, or a lean-to? I don't think I'd trade the practically idiot-proof set-up of a cat-cut tarp for the extra pitching options of a flat-tarp (which I'd likely never use). Not that a flat tarp set-up is much less than idiot-proof with practice, but with a cat tarp, I pretty much get a tight pitch in minutes w/o even trying.