Fullness of the One Who Fills

What does Paul mean in Ephesians 1:23 when he describes the church as the fullness of Christ?  Does it mean that the church is completed and filled up by Christ, or does it mean that Christ is completed and filled up by the church?

Certainly the first.  But the second is also true.  According to 1 Corinthians 12:12, “Christ” names the head-and-body totus Christus , and Christ-head without a body would be a monstrous Christ.  A Christ without a body would be a dis-embodied Christ.

It seems a perichoretic relation: Christ is filled with all the fullness of God, and fills God; we are brought into that relation of mutual indwelling, so that as we are filled with Christ in whom the fullness dwells, we also fill and complete Christ.