Why does Tomato Sauce (ketchup) require refrigeration but Barbecue sauce (like Worcestershire sauce) does not?
You don't actually have to refrigerate ketchup. Once it's opened it's good for a month or so with no appreciable change. After that the flavour and colour starts to degrade, but it's still safe to eat.

Worcestershire sauce is fermented for more than a year before it's bottled, so it will change at a much slower rate than an acidic, but unfermented sauce like ketchup. Still, it's only good for a few years after it's opened.