Tropicalesque and native: magnolia macrophylla. A similar tree is magnolia fraseri, another one is magnolia tripetala. They are deciduous magnolias, but it sounds like something you might want somewhere even if it doesn't fit the bill for the one behind the house.

Here is a link that might be useful: Magnolia macrophylla

Evergreen magnolias are also a thought - 'Brackens Brown Beauty' flowers about now, as does 'Little Gem', which is smaller - it also grows slowly. Their drawback would be the thick fleshy roots - they are close to the surface and resent disturbance, plus any shade will be HEAVY, so planting under them is not an option. Their large leaves would fit into the "tropical" look.

A Parrotia persica might be good - to 30', great fall color, scaling bark with maturity. It's deciduous, and flowers early, before the leaves.

Later flowering possibilities - all deciduous - include chitalpa (pink or white flowers, spring to fall), sourwood (white, fragrant flowers, summer, good fall color) and yellow wood (white, spring to summer, good fall color). Finding them might be a problem, but not insumountable, esp. as the last 2 are natives.

Franklinia, while deciduous as well, does flower later (July to Sept.), and has good fall foliage. Chionanthus (fringe tree) will have just finished flowering now, white, scented on the Chinese, less so on the native blooms, has yellow fall foliage. They are both findable, as I have a Franklinia and saw several of the fringe trees in Asheville Farmers Market a few weeks ago.

Paper bark maple, as well as a couple of other maples (Amur, trident, 'Bloodgood' [red leaved, tall Japanes maple]), will stay small. They won't have the late flowers, and they are deciduous, but will have good fall foliage, and winter bark interest. A gold leaved, taller JM, say 'Aureum' (?), might be good too.

How about the Daimyo oak? .... with it's large leaves I think it comes across as tropical. sorta.

Swamp Chestnut Oak (Quercus michauxii)

Thank you for the great suggestions. My personal favorite was the magnolia. However, my husband did not like it...he was awfully picky! We finally agreed on a tree and although it is not tropicalesque with large leaves and does not bloom late spring, I am happy with it. We planted a Forest Pansy Redbud. Who would have thought he would have cared as much as he did!

Forest Pansy is a great tree. I have three of them and the color of the leaves is beautiful. I also like the fact that it does not seem to create as many seed pods as the species redbud.