There is no country or official region called "Southern Ireland." The island of Ireland comprises two political entities:
"Southern Ireland" was a short-lived British administrative term (1921–1922) for the area that became the Irish Free State. It has no legal or political status today. Using it can imply the Republic is a sub-region of the UK, which is incorrect and offensive to many.
Correct terms: "Republic of Ireland," "Ireland," "the South" (informal), "the 26 counties" (historical).