Is 2-SAT with XOR-relations NP-complete?
2-SAT-with-XOR-relations can be proven NP-complete by reduction from 3-SAT.  Any 3-SAT clause $$(x_1 \lor x_2 \lor x_3)$$ can be rewritten into the equisatisfiable 2-SAT-with-XOR-relations expression $$(x_1 \lor \overline{y}) \land (y \oplus x_2 \oplus z) \land (\overline{z} \lor x_3)$$ with $y$ and $z$ as new variables.