You are not going to find what you seek, because any list small enough to be a decent answer here would be an overly restrictive diet - you need variety.
That said, if you buy:
- rice (white or brown - you'll be getting plenty of fibre so white is ok)
- lentils (not the expensive du puy ones, any other kind)
- dried beans
- vegetables that keep without refrigeration (carrots, onions, potatoes, cabbage etc)
- canned tomatoes, tomato sauce etc
- greens (eg kale) when you can - buy them and use them the same day, then don't have greens until you shop again
- canned tuna and salmon
- oatmeal (not the instant kind) for breakfast, and some dried fruit to throw into it
- some spices as your budget allows
- olive or canola oil (keeps without refrigeration) for frying the veg to start a flavourful soup, stew, or sauce
You can use these to create daily meals like this:
- breakfast: oatmeal and dried fruit
- lunch: soup with rice or noodles, lentils, and a lot of vegetables
- dinner: stew with similar ingredients to the lunch
You can vary this by making a salad of greens, carrot, cabbage, and the canned fish, by buying a loaf of bread and making sandwiches of peanut butter or tuna or salmon salad (this will require some mayonnaise) and so on.
You will be able to eat for WELL under your budget. These foods will provide protein, vitamin C, vitamin A, and fibre. You might be a little low on calcium, so drinking milk or eating cheese can take care of that, though they require refrigeration. You could go out, buy a litre of milk for a few dollars and a small (half pound?) package of cheese and make macaroni and cheese from that (drinking the excess milk) and be in a good place. (Or take a calcium supplement, or buy a milk-based drink when you go out each day.) If you can grow your own food, then you can have greens every day and do even better.
Looking for some references about what makes a healthy diet? Here's the Mayo Clinic on the Mediterranean Diet. Here's the Blue Zones guidelines on eating. You can find a lot of advice like this online.
$10 a day is actually a high budget if you are only buying food with it (no toilet paper, soap, shampoo etc) and you are not buying meat or processed foods such as canned soup or frozen dinners. For example Amazon will sell me 2 kg of rice for $5 Cdn, which is probably $4 US and this site suggests 50g per serving. Even if you double that you would get 20 servings for your $4, or 20 cents a serving. Beans? Dried beans on Amazon look like $4 Cdn a pound, and again you won't eat anywhere near a pound of dried beans per serving. And of course your local Asian supermarket is going to be cheaper than Amazon prices. So feeding just one person on $10 a day is not challenging. Eating essentially the same food day after day, having to cook for 30-60 minutes each day, having to remember to soak the beans the night before -- these things become challenging and this is why your local store sells so much frozen pizza.