How do I preserve a tomato's freshness after it is cut?

I've seen and read online, and experienced firsthand, that refrigerating a tomato is an awful idea. Not only does it lose freshness, it does little to prevent wilting and leads to flavorless tomato.

But, eventually a tomato needs to be cut if it's going to be used in slices (for say, a sandwich), and I hate to throw it out when there's still a full half of the fruit left.

How can I preserve the tomato after it has been cut, without putting it in a refrigerator and spoiling the freshness?