Red eye is caused by reflections from the back of the eye, when the iris or the eye is fairly wide. Bright ambient light causes it to close down, which reduces the amount of light entering and reflected. The red eye reduction systems in cameras fire a pre-flash with the intent of making the iris close down. This often is only partly effective.
With sunlight entering the room, the ambient light should be enough to prevent red eye.
In low light, whith the eye's iris wide open, and the flash very close to the centerline of the lens, conditions are ideal for red eye. Anything to change these conditions is going to help.