Incurvati (2020, chapter 4) gives a detailed critique of the paraconsistent approach to naive set theory. Recent work in algebra-valued models of paraconsistent set theory  gets away from naive set theory and is about placing the axioms of standard Zermelo-Fraenkel Set Theory (ZF) in a paraconsistent framework. Algebraic models along these lines are being vigorously investigated by Tarafder, Venturi, and Jockwich (see Jockwich and Venturi 2021), following Löwe and Tarafder 2015.