I have a mild case of eczema, but I've experimented a lot within a strictly vegetarian diet and have found the same results as you. A lot of foods can be triggers.
I believe in general that many agrees that
dairy
eggs
citrus fruits
contribute in eczema and that many other foods because of allergic potential can cause flare-ups. But there are many other foods who can act the same way.
I do believe all or most animal foods probably are problematic but I haven't experimented with it.
I have found that seeds in general can be problematic, also many grains, not least of which the leguminous family. Citrus fruits, coffee, broccoli and spinach can cause flare-ups. Especially raw lemon, broccoli or spinach. Cooking some foods can be very important.
Effects are often cumulative, and typically more than one factor needs to take place. (For some foods though, that means a very small amount.)
The foods that minimize any eczema or that I can eat without fearing breakouts are:
Most of the non-acidic(like citrus) and non-cabbage family fruits and vegetables. Tubers like potatoes, sweet potatoes, cassava, etc. I also eat a lot of rice, but it's not the ideal calorie source, I feel, and it's a grain. Some nuts may be OK, but they're also a kind of seed, and peanuts is a leguminous plant, so that's probably not so good. I generally don't eat nuts.
I eat a simple diet, but I get the nutrients I need.
I don't know how significant an alkaline diet is, but I think it could be important.
You could also research histamine intolerance and foods that trigger that. I think it has merit.
Good luck