I've been running mountain bikes with tubeless tires for about ten years, and have run road tubeless for about six months. I've never not used liquid sealant with tubeless, and I've always carried a spare tube. In this whole time, I've had to fix a flat during a ride, while running tubeless with sealant, a grand total of one time. It was on a mountain bike ride, and I should have known better; the tire had come off the rim during setup and had shown signs of a damaged bead. Sure enough, way out in the middle of nowhere the tire blew off the rim, sounding like a gunshot as it did. But it was no big deal, other than shaking me up a little; I had a tube and supplies to get the tire inflated again. I used very low pressure, and cautiously rode out of the woods on the damaged tire. Glad I had the tube, it would have been a long walk without it. Bottom line: carry a tube, no doubt about it.
A note about sealant and tubeless: the sealant is a significant part in getting the most out of running tubeless, IMO. For road tubeless, IMO, the main advantages are the ability to run lower pressures for better ride quality without worrying about pinch flats, and having the liquid sealant in the tubeless tire to greatly reduce flats from punctures. To not use sealant takes a big part of the advantage away.
I have communicated with Shimano about using liquid sealant in their road tubeless wheelsets (Dura Ace and Ultegra), since our shop is a Shimano dealer and this question has come up, and their official line is to not do it, there are concerns about corrosion. They say the warranty on the wheels will be voided if there are any problems relating to the use of sealant.
I'm a huge fan of Stan's Notubes products, and have built both road and mountain wheelsets for my own bikes using Stan's tubeless ready rims. We also sell a lot of their products. I'm sure there are cases of corrosion damage with various liquid sealants, but I can say I've never had a problem and find the benefits of using Stan's sealant too good to pass up. YMMV.