VIENNA (Reuters) - Social Democrats, conservatives and Greens formed a governing coalition in the Austrian province of Carinthia on Tuesday after an election rout of the far-right Freedom Party in its traditional southern stronghold.

Corruption scandals punished the anti-immigrant Freedom Party in Carinthia, where late leader Joerg Haider had built it into a national political force. Billionaire Frank Stronach's new eurosceptic party has also sapped its support.

This has clouded prospects for the Freedom Party - which opposes bailing out euro zone laggards and wants a referendum on quitting the 17-member currency bloc - of joining a national coalition after elections due by late September.

National Freedom Party leader Heinz-Christian Strache has seen his party's support drop to around 20 percent in opinion polls from 28 percent a year ago, boosting chances for the pro-Europe coalition of Social Democrats and People's Party conservatives to stay in power after this year's parliamentary elections.

Stronach's party, formed only last year, scores 10 percent or above in national polls.

The center-left Social Democrats this month ousted the Freedom Party of Carinthia - independent allies of the national Freedom Party - from power, with the rightists' share of the state vote plunging to 17 percent from 45 percent in 2009.

It was the Freedom Party's worst showing in the province since Haider, who died in a 2008 car crash, took charge of the nationalist party there in 1983.

The next provincial elections take place in Tyrol on April 28 and in Salzburg on May 5. National elections are tentatively planned for September 29.

(Reporting by Michael Shields; editing by Ron Askew)