Rome can be considered an early attempt at a "globalized" society - while not strictly global it did bring many different regions and cultures over a very wide area under one administrative umbrella, and opened the doors of trade. Among other things, that lets you do things like regularly consume dishes that use geographically distant ingredients. You could probably go down to the store and buy all kinds of spices pretty much anywhere in the Roman Empire.
I think you're underestimating what could be done with preservation technology such as salting, curing, drying, pickling, fermenting etc. Meats were in fact frequently traded. Salted pork was a staple food. Cheese even more so.
And what of liquids? You're surely aware of ancient wrecks filled with amphoras? What was in them? Surely not cereals. Oils, wines, vinegars, acidic sauces such as the famed "garum" - all travel well.
