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.

There is no definitive way to answer this without knowing the specific formulations of the products in question, which is almost certainly proprietary to the producing company.

It is likely that the so-called barbecue sauce is in fact more than tomato sauce mixed with Worcestershire; it may be more acidic than the the tomato sauce, giving it greater stability at room temperature.

The added acidicy that Worcestershire sauce and Barbecue sauce both have ensures that they don't go bad nearly as soon as tomato juice does.

Worcestershire sauce is actually the result of an attempt to pickle a few ingredients gone awry.

Further, on a side note (the tomato juice reminded me of this) a great technique that Jeffrey Morgenthaler uses to ensure that his bar can serve bloody mary's easily and efficiently is to mix all of the acidic ingredients together and make a mix that can be stored for about 2 weeks at a time and mix in the other ingredients as the bloody mary is made.

http://www.jeffreymorgenthaler.com/2013/the-bloody-mary-conundrum/

The answer is simple - tomato ketchup does not need to be refrigerated provided it is in a glass bottle - it's only the ones in plastic which advise they need to be kept in the fridge after opening. Quite why that is I've never bothered to check, must be something to do with the material of the container. I've never seen Worcester sauce in anything other than a glass bottle, but I assume, were it in plastic, it might also need refrigerating.