Carthage had a reputation as a mercantile powerhouse. Did they have any "signature" products, i.e. were there any goods that if you thought of it, you'd think "ah, that's from Carthage"?

It's funny you should ask that, because I was just reading some collected works on the topic of the Vandal Conquest of North Africa which was brought on by watching the excellent (in my opinion) documentary on the Dark Ages, called the Dark Ages: an Age of Light.

Carthage, and North Africa in general was a major agricultural center for the Roman Empire. Perhaps owing to this fact, there is a very specific, well studied form of pottery called African Red Slip Ware (ARS) that travelled throughout the empire. Because pottery is relatively durable and ubiquitous, ARS is a highly studied product. Since its production is centered only in North Africa, its presence is used to measure the health of trading patterns of the Roman Empire. ARS showed up in areas as far flung as Britain.

ARS is seen throughout the empire even during the times of the Vandal Conquest. Which is interesting, given the amount of hand wringing in the Empire over the Vandals. One of the few times through the late roman period where the east and west collaborated on closely was the reconquest of North Africa. The very name Vandal today is associated with crime and destruction of art, yet even during the height of the Vandal Empire art and trade thrived.

Sources: a well regarded source on this topic is John W. Hayes 1972 work called Late Roman Pottery, but I do recommend everyone watch the BBC series The Dark Ages, and Age of Light for an entertaining primer on some of the groups that lived in that period. In my opinion, today our views of the Germanic Tribes, the ERE, and the Carolingians are skewed by later historians.