Broad and Cummins had another back-and-forth shortly afterward asking him: 'Do you think that's out?' after the Australia captain dropped a difficult caught and bowl chance off Ben Stokes.
After evading a bouncer, Broad deliberately grounded his bat behind the crease and waited for Carey to pass him before moving.
Starting the day at four for 114 and needing 371 runs for victory, Ben Duckett and Stokes made a strong start, with the England captain surpassing 50.
Duckett was eventually dismissed for a well-crafted 83 when Carey took an outstanding one-handed catch down the legside.
Stokes reached his century in just 142 deliveries as England continued their improbable chase.
Meanwhile, Usman Khawaja and David Warner engaged in a confrontation with members of the Marylebone Cricket Club in Lord's Long Room.
Khawaja was seen arguing with a club member on his way back to the dressing room before being pulled away by a teammate and staff member.
Boos reverberated throughout the room as the Australian players walked past the MCC members.
Image source: Reuters, ICC
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