First, an "EAI tool" is a pretty generic thing. It's not an acronym I find particularly useful, really, because it can be so incredibly broad. Essentially, anything that acts as a connector to promote sharing of disparate data (usually in real time) can be considered "EAI" in one form or another. BEA's Tuxedo is used to manage complex transactions that have to be handled with a high level of both accuracy and integrity. An e-commerce site might be built with Tuxedo playing to role of transaction manager to ensure that a transfer of funds is completed on both sides -- what's called a "two-phase commit" -- when perhaps both ends of the transaction aren't in the system. That's just one simple example, but from that standpoint I suppose you could classify it as an EAI.

I suppose you could manage those transactions with "messaging," but I'd really best refer the question for any more depth to someone specializing in transaction processing. Tuxedo is much closer to IBM MQ Series than to a human messaging product, where I spend most of my time.