According to the Daily Mail, Richard III will run from 14 July 2012 with Twelfth Night opening on 22 September. It will mark Rylance's first appearance on the Globe stage since he stepped down in 2005.
The actor, currently reprising his Olivier Award-winning role as Johnny 'Rooster' Byron in Jerusalem in the West End, will be directed by Tim Carroll in both productions. Carroll directed him in several productions during his ten-season tenure in charge of the Globe. Both Twelfth Night and Richard III will be staged with an all-male ensemble, according to original Shakespearean practice.
The Mail also reports that Rylance hopes to transfer the productions to New York care of producer Sonia Friedman, though nothing is yet confirmed.
As previously announced, the Globe will stage an epic Globe to Globe season next April, featuring all 37 Shakespeare plays being performed in 37 different languages. The full season will then commence on 7 June with Jamie Parker playing Henry V, directed by current artistic director Dominic Dromgoole.