You already almost have an air barrier. You could simply tape the joints with appropriate tape.
From the Owens-Corning product page for their tape for their foam (no endorsement implied, just as an example of the product class existing for the application):
HomeSealR™ Foam Joint Tape is an effective product to tape the joints
of FOAMULAR® extruded polystyrene (XPS) board in residential vertical
wall applications when it is used as continuous insulation over steel
or wood wall framing. HomeSealR™ Foam Joint Tape provides reliable
adhesion to FOAMULAR® XPS insulation. The durable backing is coated
with an aggressive acrylic adhesive to offer excellent adhesion
coupled with strength and is easy to handle.