I live in the area where Tyson has most of its poultry operations, and I know a few people who work for them fairly well - including a couple who are working there under false identities due to immigration status.
Tyson knows. They couldn't _not_ know. They seem to maintain an acceptable level of "plausible deniability" at least, and I've not seen any evidence that they're explicitly allowing it to happen, or even that it's their preference. It seems more like they _have_ to tolerate it. If they tried to aggressively stamp out their workers using false documentation, they'd lose a big part of their experienced workforce and would incur significant additional labor costs as the market adjusted to the change.
More to the point, those people aren't just going to shrug their shoulders and head back to their home countries. Many of them would just go through more complicated steps to get false documents that passed the increased scrutiny. Tyson would have to pay more to get workers. Much of that money would be spent by the very same employees obtaining and maintaining false identities, and the people they're getting them from aren't exactly upstanding citizens.
If anything, I'd expect a crackdown on illegal labor practices here to result in funneling more money to criminal organizations (i.e., "the cartels") and tying the immigrant community more deeply with them. It really looks like a lose/lose proposition from where I sit.
NOTE: None of the above is a comment on US immigration policy, or indicative of my personal position on how that should be handled.