And what support test would that be?  Who is paying more than half the household expenses?  It's possible to have more than one household under one roof, but I haven't seen it work in a situation like this, where both kids belong to both parents.  

The low-income non-spouse doesn't need a dependent to reduce taxable income to zero, but may benefit from claiming one or both kids for EIC.  Doesn't need HH for that.