As far as I can tell, ketogenic diets solve the problem of fat loss. I know, anecdotes are not data, but it's worked wonders for everyone I know who's tried it (myself included).

Err on the side of posting solutions which may not be universal but are still likely to be helpful to many people.

This is the sole reason I'm posting this. Keto works for very many people. The short story of keto is that your brain can only eat certain kinds of chemicals. Glycogen from eating carbohydrates is one of them. Ketones generated from fat is another. Your body will preferentially use the first over the second, since turning fat into ketones is expensive. So if you eat few enough carbs (<30g per day is the figure I remember) and plenty enough fat (2:1 fat to protein is what I heard), your body will eventually start doing chemistry that turns dietary and body fat into ketones.

There's some more practical advice about how to induce ketosis quickly (muscles store glycogen, so exercise helps) and how to make low-carb versions of foods you enjoy, but that's pretty much the gist of it.