Answer: Ivy can be terribly invasive, as you've discovered. Have you considered growing grapevines? They may not produce grapes, but they thrive in full sunshine and are quite attractive in their own right. Other suggestions include:
Clematis. I can't recommend just one. The red, purple, pink and white five- to six-inch flowers of the jackman clematis that bloom profusely in late spring are one reason to choose this vine. The anemone clematis ( C. montana) grows vigorously to cascade over fences, walls or trellises and is covered with white to pink flowers in May. Another is sweet autumn clematis. It will run rampant over any plant or structure in its way and can become almost weedlike, but doesn't need a heavy support. The lacy white, fragrant flowers open in fall. Prune hard after flowering.
Climbing rose. There are so many hybrids of climbing roses, in miniature and full size forms, that to recommend one or two is ludicrous. Having said that, there are two we have grown for years. One is planted between two structures, receives about six hours of light and has totally taken care of itself. It is called aloha. Another one we have enjoyed is zepherine drouhin and is completely thorn-free. It has performed admirably on benign neglect. At least two flushes of pink flowers are produced per season on both.
Everlasting or perennial sweet pea. Flowers of deep pink to purple, summer and fall, are a good reason to grow this fully hardy vine, which will cover a six-foot trellis in a season. Wide winged stems are an interesting growth characteristic. Freezes back in winter.
Native or Japanese wisteria. This woody twining deciduous vine has lavender, fragrant flowers in spring. It is pleasing in full bloom, trained on pillars, a deck, stair railings or another sturdy structure. It needs heavy supports, such as pipe or lumber, not lattice. Training is a must. It grows strong woody stems around everything it touches and seeds everywhere, sun or shade. Try less invasive native Kentucky wisteria ( Wisteria macrostachys) or American wisteria ( W. frutescens).
Purple lablab or hyacinth bean. A fast-growing vine ( Dolichos lablab) that starts from seed annually. Deep pink flowers and red pods of this tendril climber are outstanding. Grow in a planter on narrow stakes, a fence or lattice in full sun. Let some go to seed to save and grow next year.
Best wishes with your landscape.
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