Blend the chili peppers and garlic. In a blender or a food processor, add the chopped red chili peppers and chopped garlic, and process to a fine paste. Alternatively, you can chop them very finely by hand, then mix in enough water to make a paste. Set the fine red chili and garlic paste aside.
Heat vegetable oil. Add vegetable oil to a large saucepan, preferably cast-iron, and heat oil on medium to high heat.
Cook paste. Once the oil is just hot, add the fine red chili and garlic paste. You know the oil is just hot when it begins shimmering. You do not want your oil to be too hot because the paste could burn. Cook until the ingredients smell slightly fragrant, about 5 minutes. Stir a couple of times. Add the ¼ cup of water to the pan and continue cooking the paste until the water evaporates a little bit.
Add the lemon juice, rum and soy sauce. When the water has evaporated a little bit, add the lemon juice and rum. Bring the mixture to a gentle boil, just enough for the liquid and alcohol to evaporate and mix everything well and reduce heat to medium. Add in the soy sauce, and optional fish sauce if desired. Cook another 5 minutes. Taste and adjust the soy sauce and/or lemon juice to your liking.
Serve. Enjoy this delicious Homestyle red chili paste with any dish of your choice, especially Chinese and dumplings!
Storage. To store, let the paste come to room temperature before transferring into a container. It keeps well in the fridge for several days and the flavor will develop over time. You can top it with some vegetable oil to make it last for a few weeks.