MINNEAPOLIS (AP) _ Republic Airlines Inc. said Tuesday it lost $10 million in the fourth quarter, compared with earnings of $4 million a year earlier.

But Republic earned a record $29.5 million for all of 1984, rebounding from a $111 million loss in 1983.

It also was Republic's first full-year profit since 1979, when the company was formed through the merger of North Central Airlines and Southern Airways. Republic also acquired Hughes Airwest in 1980, and between 1980 and 1983, Republic lost $222 million.

Republic said its loss in the latest quarter reflected one-time expenses and an interim decline in traffic associated with a restructuring of its routes.

Stephen M. Wolf, president and chief operating officer, said in a statement that the 1984 profit ''reflects Republic's emphasis on route realignment, lowered operating costs and employee commitment to the company's Partnership Plan, which saves the company $100 million annually through 1986.

''Employees have been responsive to many changes in the past year and their dedication and willingness to meet new challenges has been crucial to Republic's 1984 financial improvement,'' Wolf said.

Republic's unions agreed last year to extend a wage-concession plan through 1986, a plan that includes a 15 percent wage cut and additional cost savings of 8 percent through productivity gains.

In return, union members received 5.5 million shares of common stock, or about 15 percent of the airline's common shares; $100 million in profit sharing over several years and representation on the company's board of directors.

The fourth-quarter loss came on revenue of $357.6 million. A year earlier, Republic earned 10 cents a share on revenue of $348.3 million.

For all of 1984, net income equaled 75 cents a share. Annual revenue climbed to $1.55 billion from $1.51 billion.

In 1984, Republic began service to eight new destinations while suspending operations to another 23 locations. The route realignment focused the company's resources on its three main connecting centers, or ''hubs,'' in Detroit, Minneapolis-St. Paul and Memphis.

''This route restructuring will continue through 1985 as Republic positions itself for continued expansion into markets that will contribute to the development of the three hubs,'' Wolf said.