Do you know this (applicable to Illy, too)?
Otherwise, a good place to start.
This is a common problem that starts with calibrating your monitor, synching your applications, and calibrating your desktop printer ( in your case, the Efi RIP / Epson 7800 combo ). To get you where you need to be would take a few minutes and would require us to know what your current color settings are. But, very often prints do not match the monitor because of the two completely different color spaces ( transmissive vs. reflective ) and their corresponding color gamuts. Follow Jacob's advice and do some research, starting with your RIP's User's Manual, your printer's User's Manual, and some Adobe White papers on Color Workflows ( adobe.com/whitepapers ).
Followup. Try printing with color management turn-on in the RIP and see if that helps. It gets a bit technical, but most desktop printers need either a driver or a RIP to convert the file's colors to printable colors from the printer. This conversion is done in the RIP ( in your case, the Efi ) where the printer will print predictable color that it is capable of. A few years back and maybe even today, that requires your Epson to utilize color tables from the RIP to produce accurate color. So, if you choose "Have Illustrator decide color" and the ray percentages are sent to the RIP without maximizing color management in the RIP, you may get some hues and values a bit out of gamut.