I try to avoid getting into that situation. In fact, I might have encountered a situation like that maybe once or twice in all the years I've been cycling.

I'm not a hill climber, so I try to avoid hills whenever possible. Even more so on the tandem. We simply do not ride the tandem in really hilly terrain. I'm just not comfortable with that ... going up or down.

If I am climbing (with a single bicycle), I prefer not to stop unless I'm planning to get off and walk. For the most part, I'd rather just get into a comfortable gear and keep going to the top. Or if I am going to stop, I pick my moments and stop on a flatter bit of the climb or a roadside turnout or something. And I haven't encountered a lot of stop signs halfway up the hills I climb.

If I do find myself in the rare situation where I stop and then attempt to get going again, I do the thing where you point the bicycle across the road to get a bit of a coast going and then swing it around to continue climbing.