Some of the passion flowers are evergreen, I have a constance elliot in my garden and its lovely also Star Jasmine is a lovely climbing plant too, if you want something fast buy a well established one.

Trachelospermum jasminoides fits the bill (except its not particularly fast growing) - but only if you are planting it in a sunny, very sheltered spot. I can't think of another evergreen climber which flowers in the summer, other than Jasminum officinale, which is only semi evergreen, and gets enormous. If the space you have available is less than 10 feet wide by 30 feet high, I wouldn't recommend it.
There's a new honeysuckle out which, if you believe the sellers, is evergreen and flowers from spring through till late summer - Lonicera japonica 'Dart's World'. Whether it actually lives up to the sales hype is another matter though.

Winter Jasmines (a common name for Trachelospermum jasminoides) and Lonicera japonica Halliana are my suggestions, to expand on what Marion and Bamboo have mentioned. English and Irish Ivies are not to be overlooked either and you can have several climbers next to each other.

Climbing roses. With Kildermoriies ivies could look pretty

Honeysuckle Halliana fits the bill

Here it is in my garden

http://www.flickr.com/photos/31559373@N00/4760266453/

and more details here

http://apps.rhs.org.uk/plantselector/plant?plantid=1187

Clematis armandi is a good one.

There are a few roses that will fit the bill also.

Evergreen roses?

Ha ha, anchorman - lots of the plants mentioned don't fit the parameters in the question, which were, evergreen, SUMMER flowering and fast growing.

Try Lonicera halliana. An evergreen honeysuckle that is a climber but not self clinging. As long as it is not in a totally exposed area it should be fine. I moved mine from N. Beds when I moved and it's still going fine.