The AL Central champions dealt for Romine a day after they learned Iglesias could miss the season because of stress fractures in both legs.
''We talked about having a shortstop you can catch the ball and run a little bit,'' general manager Dave Dombrowski said. ''He fits that description.''
Former Gold Glove shortstop Omar Vizuel, now a Detroit coach, was with the Angels last season and put in a good word on Romine.
Romine, 28, played in 47 games for the Angels last season and batted .259. He is considered a good defensive player and stole 15 bases in 89 minor league games last year.
''It's all a plus if you have a little speed down at the end of the lineup,'' Ausmus said.
Romine made his major league debut with the Angels in 2010 and has a .250 average in 74 career games. The fifth-round draft pick in 2007 is the son of former major league outfielder Kevin Romine and the brother of New York Yankees catcher Austin Romine.
Perez made his major league debut last year and Worth has been a utilityman with the Tigers the past four seasons.
The 24-year-old Alvarez pitched in 14 games for the Tigers last season, making six starts. He was 1-5 with a 5.82 ERA.
Alvarez split last season between Detroit and Triple-A Toledo. He was 8-6 with a 2.80 ERA for the Mud Hens and named Tigers minor league pitcher of the year.