It will mark the first time any play has been live streamed from the venue.
The broadcast will be available to online audiences for free on the Shakespeare Lives website and via the BBC iPlayer on 11 September.
The production, which opened in April, is Emma Rice's first since becoming the Globe's artistic director.
Rice said: "I'm thrilled that my first production at the Globe is being filmed in this innovative and energetic way."
The play received warm reviews from critics when it opened at the theatre, located on London's South Bank.
The live stream, which will be introduced by actress and comedian Meera Syal, will mark the final night of the play's residency at the Globe.
Shakespeare Lives is an online digital festival co-curated by the BBC and the British Council, running all year to celebrate 400 years since the playwright's death.
Cortina Butler, the director of the project, said: "We're delighted to play a part in making it possible for a truly international audience to experience a magical, modern production from Shakespeare's Globe."
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