Unusual three-way+ light configuration
Real simple.  2 of the fluorescent lights are on the 3-way switches.  

Separate from that, 2 of the fluorescent lights are on the 2-way switch. 

The reason is to give a "low-high" setting. 

The wire going from the 3-way switch to the 2-way switch is always-hot supply power.  The other 3-way switch is the "remote".