Two evergreen vines I have, here in a zone 7b garden, are Gelsemium sempervirens and Clematis armandi. Have no idea if they would do well with you. The Gelsemium is extremely vigorous. The Clematis only arrived this Spring but is growing beautifully. I do not know if it will pull through a Winter, here.
I think the hazard with Akebia is not merely seedlings, but it's ability to run underground and sprout many feet from where the desired plant is growing. Mine is only two years old, so has not done so, yet.

Thanks bogturtle, I'll look into those. We are a bit colder than you but I can mulch and always like pushing those zone boundaries.

I've had my akebia for 5 years and have seen no sprouts elsewhere.

Clematis armandii would definitely be my first choice. It prefers part shade. Avoid afternoon sun! I grew one on a fence in my garden near Dulles airport in Northern Virginia. It would die back somewhat in a really harsh winter but it survived single digit temperatures and was still thriving after eight years in a raised bed beneath very tall tulip poplars.

Here is a link that might be useful: Clematis armandii

I'm not sure but I think Dallas, Texas is in zone 7.
I want to plant an evergreen vigorous perennial climbing vine.
The brick wall does not get afternoon sun.
Vine suggestions will be appreciated.
Carolyn

Squirrelspur,
A bignonia vine should meet your needs. Crossvine is another name for it, and either Dragon Lady or Tangerine Beauty should be available in your area. cwc000 in Dallas, same for you. These stayed evergreen for me in zone 6, and they bloom at least twice a summer. Not invasive here, but I pull them up off the ground and let them wrap around a deck.
Cindi