I think if you follow LCHF properly, then you will naturally tend towards healthier food anyway. Firstly we are advised to dump the junk and processed foods. We are encouraged to eat above ground veg rather than root veg, We will tend to eat more salads and things like coleslaw and tomatoes. I have regularly added avacados and olives. We increase our intake of nuts, and berries
It is just the reduction of fructose containing foods that is a potential problem, but then again I tend to use Ceasar Salads and apple/cheese/mayo salads on the side. I believe some fruit is acceptable with care, but the rule I follow is reduce the tropicals, and avoid fully ripe or over-ripe fruits. Then its simple.
As someone on an iron supplement for anaemia, I am myself adding Vit c supplements anyway, and its no big sweat.
I cannot say about other LC diets, such as Atkins, Paleo etc