I have an Eagle One Vertical and ICOM 7100. Your RT-100 is designed to be mounted right at the base of the Eagle One. Hopefully that is the way you have it installed, since putting it on the end of the coax cable that is on the transceiver end is not going to work well... if at all. You'll read replies here saying you need to put an antenna analyzer on the Eagle One first before you attach any tuner. Wouldn't hurt, but I can tell you the SWR will be off the charts on all bands except 15M and 40M... where it will be passable but not great. However, you will be able to tune those two bands... and all the others... with the tuner. So now a question: When you are seeing "off the charts" SWR, where are you measuring the SWR and how? The reason I ask is to help determine if there is a defect in your RT-100... which is unlikely. But does the tuner allow your transmitter to operate? And on which bands will it work / not work?
FYI - I've heard that whenever the makers of the Eagle One are at a show with their antenna, they have an ICOM AH-4 tuner set up at the base of the Eagle One. After much experimenting with my Eagle One and IT-100, I've been able to improve the match by adding a Palomar 4:1 impedance transformer at the base of the antenna. But I'm convinced that the ICOM AH-4 would be a superior solution. However, I'm also surprised that your new RT-100 is not working similarly to an AH-4 (assuming you've installed the RT-100 as specified, and have checked to ensure you have no coaxial or connector "opens" or "shorts").