I would like to know which companies actually manufacture the t-shirts
for the following brands:

Polo (Ralph Lauren)
Nike
Tommy Bahama
Hanes

In other words, I know that Nike doesn't actually "make" their own
t-shirts, but rather gives their specs and standards to an outside
company which is then contracted to do the work.  I'm looking for that
company.

Also, please note I am looking only for the t-shirt manufacturers for
the above brands (I'm aware they may contract with different companies
for their various other product lines).

I know a guy who used to be involved in this.

The way it works is that the buyers use a Western intermediary who
places the contracts with local manufacturers.
The reason for the intermediary is that they know the local ropes,
dealing direct with the manufacturer would be ... well frustrating.

The company he worked for was pretty large, its head office was in
Hong Kong, but it operated in the Philipines, India, Thailand, PRC -
you name it.