How do complementary doctors treat diverticular disease?

Variably. The condition requires only simple medical and dietary recommendations. Diverticulitis is a complication, and treatment is different depending on severity.
Poorly. They may suggest alterations in diet which may aid in limiting the development of diverticulitis, and in easing the stress produced on the colon wall by very hard stools. They may also use stool softeners, but these are therapies used by traditional physicians as well. If you have diverticulitis I would suggest not relying upon "natural" attempts at cure. This may be detrimental to your health.