Historically, proofs of the Boolean Prime Ideal Theorem (BPI) in choiceless models of ZF have taken considerable effort. We present a new conceptual framework with which to prove BPI in choiceless models, based on Harrington's proof of the Halpern-Läuchli theorem. This allows for new proofs of several results from the literature, including the fact that ZF+BPI cannot prove the axiom of choice. We describe these results and the connection to the Halpern-Läuchli theorem that is implicit in this approach.