I have Zephirine Drouhin, a shade tolerant, nearly thornless climbing rose with Prince Charles clematis, a light lavender-blue. They don't have a completely overlapping bloom, with ZD beginning first and PC overlapping with the later ZD bloom. Here it is in late June with PC just starting. I have a trellis because ZD and most other climbing roses aren't reliably cane hardy here, so I've never tried this without some supplemental support.

You can grow clematis into any stiff-stemmed shrub. One of my favorites is Clematis Little Bas growing into Hydrangea paniculata 'Quickfire'. Here, the clematis starts blooming mid-June with the GF buds, then the bloom overlaps for a couple of weeks, and then the QF continues blooming until frost with the blooms shading slowly to deep pink.

Clematis Mikelite and Romatika scrambling around on top of a row of Fothergilla shrubs. The Fothergilla has early spring bloom and stunning autumn foliage color, so this gives it some summer interest.

I was inspired to do this a number of years ago by another gardener on this forum, so I am just passing on the enabling.