I have a Clematis Montana 'Elizabeth' which has white flowers. It grows strongly in their second year and flowers well (when not knocked about by wind and rain of this summer). Hardy Jasmine is also white and is evergreen, so mixing the 2 in would be a good match. Both are not self clinging so would need support on a brick arch.

Self clinging climbers with white flowers...only one I would say is a climbing Hydrangea - Mexican climbing Hydrangea. Only thing I know about that is that it's deciduous.

I like those ideas kilder, ive montana and is very reliable

I'd ask whether the arch is in sun or shade, and also how much room there is for a climber in terms of height and spread....
Not sure which Jasmine you mean Kildermorie - Jasmine officinale, fragrant white flowers in summer, isn't evergreen, but perhaps you mean Trachelospermum jasminoides, which is evergreen? Needs sun and a relatively sheltered position.

I have an arch with a summer jasmine one side , and the yellow winter jasmine the other side, so have flowers most of the time.

I meant Trachelospermum jasminoides - which I think is called winter Jasmine. Jasmine officinale is not terribly hardy I found, it at least tends to lose all its foliage in a wet winter.

How about a white clematis? Or a white rose such as climbing iceberg?

Clematis montana? I have a lovely pink one, Superba

I dont seem to have much luck with Clematis, grows each year to about 2 foot then dies back until next year.
other ideas answered dont seem to be climbers but bushes, my brick arch is one brick thick and is about 7 ft high, could send a photo of arch if required.
Brian.

Rosa 'Rambling Rector'. Grows a long way but can be cut back after flowering. Plant a, or maybe more, Clematis to grow up it.

Montana is very vigorous and mine has never died back....more a matter of cutting it back! Small flowers but absolutly covered and we,re on a windy hill open to the north and east.......

Really really vigorous (nearly uncontrolable) is the russian vine which is creamy white......i love my virginia creeper and boston ivy (parthenocissus) stunning wintercolour and a framework for winter jasmine
Pyracantha has creamy flowers followed by red or orange berries which the birds love, not self clinging, then rambling roses, again not clinging but mixed with things that are....
You will probably need a frame work of wires through 'vine eyes' anyway whatever you grow or trellis