For the ackee and mushrooms, heat the oil in a heavy-based pan over a medium heat. Add the sliced onion and peppers and fry for 4-5 minutes, or until softened, stirring regularly.
Add the chopped tomato and continue to fry for 1-2 minutes, stirring occasionally, or until the tomatoes have started to break down.
Add the mushrooms and continue to fry for 2-3 minutes, or until the liquid released by the mushrooms has evaporated.
Add boiling water, as necessary, to loosen the sauce mixture, stirring until well combined in the mixture. Season, to taste, with salt and freshly ground black pepper.
Add the ackee and carefully fold it into the sauce. Simmer for 2-3 minutes, or until heated through, then sprinkle over the spring onion. Remove from the heat and keep warm.
For the steak, rub one teaspoon of the oil all over the pieces of fillet steak. Heat a heavy-based pan over a medium heat until hot, then add the steak pieces and fry for 1-2 minutes on each side (for medium-rare), or until cooked to your liking. Remove from the heat and set aside.
Meanwhile, heat the remaining half a teaspoon of oil in a separate, heavy-based pan over a medium heat. Add the onion and sweet pepper and fry, stirring regularly, until softened.
Add the remaining steak ingredients and continue to fry for 4-5 minutes, stirring regularly, until the vegetables are just cooked through. Season, to taste, with salt and freshly ground black pepper. (NB: Bear in mind that the soy sauce is quite salty.)
Add the cooked steaks to the pan and cook until heated through.
To serve, place two pieces of fillet steak onto each of four serving plates. Pour over the pepper and tomato sauce. Spoon the ackee and mushrooms alongside.