I would paint it just so you could see it! Are you facing south?

wood will always need constant upkeep if exposed to sun. A good marine varnish will work, but you still have to apply a coat every year or two.

If you decide on paint, pick a nice, bold color that compliments your color scheme. A glossy eggplant or sage green would look nice.

something along this tone, but darker

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Sun exposure is a booger to deal with. I agree with the suggestion to paint the door. If possible, take the door to an auto body shop for a pro sanding and enamel spray paint job. You will have to do the sidelights in place. beth's suggestion of eggplant would be a beautiful color for your lovely home.

Between the heat and humidity, I think you'd be better off with a steel/fiberglass door. You can paint it a dark expresso color to mimic wood, or any other color.