Our Highlander and EX trim Accords have the map lights turn on with the dome light when the door opens. My LX SE does not. This workaround did it for me and took all of five minutes three feet of wire and a coat hanger:
Basically you run a wire under the headliner from a vacant port on the harness of the map lights to the ground of the bulb socket on the dome light.
Originally did this to convert to white LED's. Bought a couple that were an annoying blue (maybe good for some people) and some that weren't bright enough. Found some that I like on e bay, 200479826030 with 8 surface mount leds, that are brighter than the stock 3175's and white as well. So maybe I didn't need to do this after all, but thought someone else might find it helpful.