Use 110 volt lamps connected in series. One in the cloak room and one outside. would solve the problem. A long as they were both the same wattage they would be equally bright.
With careful and informed selection of lamps it was possible to have 230 volt 60 watt lamp in series with a 12 volt 3 watt lamp as both took the same current BUT do NOT try this in the domestic situation. Also the two lamps have to have the right cold resistance to prevent the 12 volt lamp burning out on the inrush current when they are turned on and the filaments are cold.
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