Their home ballpark was Al Lopez Field, and they were a member of the Class A Florida State League (FSL) from 1957 until 1988, mostly as an affiliate of the Cincinnati Reds.

The city built Al Lopez Field in 1954, and the Tampa Tarpons became the ballpark's first tenant when they began play there in 1955 as a new member of the Class D Florida State League.

That season the club, renamed the Tampa White Sox, drew 55,900 fans, seventh in the 14-team FSL, and went 35–35 in the first half before they won the western division second-half title at 36–24.

[7] Other Reds players from that era who started with the Tarpons include Ken Griffey, Sr., Johnny Bench, Dan Driessen, Rawly Eastwick, and Dave Concepcion.

When the Reds won their next world series in 1990 with manager (and Tampa native) Lou Piniella, they again had several former Tarpons on the roster, including Tom Browning, Rob Dibble, and Paul O'Neill.

Another notable Tarpon was Randy Poffo, whose brief minor league baseball career ended in 1974 after batting .232 as a 21-year-old outfielder.