Reducing 3SAT to a Set Splitting Problem
Your problem is a generalization of NAE-SAT, an NP-complete version of SAT (see for example David Witmer's lecture notes in which we want each clause to have at least one satisfied literal and at least one falsified literal.

In order to convert NAE-SAT to your problem, identify literals with items, have a set per clause, and also have a set per variable, forcing the two literals to get opposite colors. Details left to you.