Alice understands Bob’s perspective by translating them into the shared language of a 3rd-person perspective. If Alice wants to judge the truth of Bob’s beliefs, she can translate her beliefs to the 3rd-person perspective as well. Once everything is in the 3rd-person perspective, correspondence becomes simple equality.
But this only covers agreement between Alice and Bob, not that Bob’s beliefs correspond to reality.
You say:
But this only covers agreement between Alice and Bob, not that Bob’s beliefs correspond to reality.