Let's say the White Sox move to Tampa in 1989, as almost happened. Frank Thomas comes up in '90 as per OTL, Jerry Reinsdorf keeps the team a couple years and the GM stays on...
Thomas would be a free agent after 1996, presumign he doesn't sign a long term deal. Would he want to stay there in Tampa? Remember that Andre Dawson left Montreal due to the turf doing a number on his knees, though Thomas played first so maybe that's not as vital.
More important is that the White Sox, while they don't draw as badly as OTL's Rays as first, probably do drop 100,000 to 200,000 in attendance from OTL's Chicago, and also, Vince Naimoli might not hve the money to keep as many big name players.
On the other hand, they probably don't make the Albert Belle signing in '97. Perhaps that money is used to sign Thomas. Belle went to Chicago in part, he claimed, becuase it was Cleveland's rival - so maybe he goes to the Yankees or somethign instead.
Where would Thomas go as a free agent? My first guess is the Angels - J.T. Snow was leaving after '96 OTL, and they did poorly at DH in '97 so even with Erstad at first they could put Thomas at DH. It seems like a good fit. Or, would Tampa trade him after a disastrous '95, not knowing if they could re-sign him?
Having Thomas in the N.L.would be interesting, too. He couldn't be a DH - but he might survive at 1B a few seasons. Perhaps an N.L. teams signs him to a 2-3 year deal, and he figures he can still milk his power and a free agent contract again, for more money, after 1998 or 1999. After all, he's still be fairly young.