Drink plenty of water, keep moving (as in walking) and don't neglect the "eating fat" part of low-carbing. I have had no problems with constipation since starting low/no carbs several weeks ago.
By the way, if fibre is the "natural cure for constipation"... does that mean all carnivores are constantly constipated

What about the Inuit, or many Native American tribes, or the Masai, or many other traditional communities that rarely, if ever eat/ate fruit and veggies

And who says that diet has to be "well balanced"... is it the same "experts" who say we need to eat a high-carb diet?