It's a genuine dilemma. Where I live, violent thunderstorms can be common in summer, with high winds and lashing rain (or hail!) plus crazy high temperatures. It's not quite as common in eastern Ontario, where I live now as it is in southern Ontario, where I grew up - but it happens.
To be honest, a lot of times people just try to stay inside while this is going on. The good news is that these storms come and tend to blow over in half an hour or so.
Anyway...it does make dressing for real life quite difficult. I tend to carry a small collapsible umbrella. As Ginger says, in the wind, this doesn't exactly work, but it helps a bit, and can also help to keep feet dry. I angle the umbrella so as to receive the greatest protection.
A raincoat is far too hot, so I tend to wear clothing that 1). won't feel too revealing if it gets wet (so darker colours) and 2). can dry relatively quickly. I bring a faux-leather bag or my Saffiano leather one (which stands up to the rain really well.)
I agree with Adela that footwear is the worst! I hate the feeling of feet sloshing in soaking wet sandals or shoes! I try not to wear suede, and instead choose a pair of older shoes or sandals, as a rule. Sometimes I'll wear my Fitflops. The tops are leather but the thick rubber soles help keep me out of puddles at least.
