Reika Mishima is a mystery, probably the biggest mystery in the anime. We're never given a straight answer as to who she is, but all the clues and hints are definitely in the series.
She's directly connected to the RahXephon, and also connected to the heart of the instrumentalist (pilot). You could say she's the personification of the giant robot, its manifested soul, in essence, when you see Reika Mishima, or when you hear her speak, you can be certain that you're hearing the RahXephon speak. The word the anime uses to refer to Reika Mishima's true persona is Ixtli, which literally means "true face."
Ixtli can take any shape or form it wants, however she constricts herself to take the image of the person in which the Instrumentalist desires deep in his or her heart the most. Why? Because Ixtli needs to establish a close connection with the pilot in order for the RahXephon to function; Ixtli needs to be wanted by the pilot, if it isn't, the RahXephon will simply revert back to its egg form. It's simply easier for both the pilot and the soul of Xephon to be intimate if Ixtli takes the shape of somebody they love. We see this very clearly when Ixtli started to fade and the RahXephon reverting back to its egg form late in the series because Ayato was confused and didn't really want anything to do with Xephon anymore.
Reika is also the protector of Ayato. On several occasions she rescued him from trouble such as in case when Ayato's mind was taken by the Dolem around midway through the series. She's the one that got Ayato out of that delusional world. And also when his hands were very cold during the time he was fighting Sayoko's dolem she gave him a pair of mittens... the same ones Haruka gave him earlier in that episode.
At the end of the series, we see Ayato drawing a canvas of a girl looking at the sea with the signature yellow dress. When he asked for her name, she said she was Haruka... Haruka Mishima. If you didn't go "WTF", that means you paid attention in the series. Good! If you did... well, read on.
Who does Ayato love? I think it's fairly obvious to say that it's Haruka he loves. However, throughout most of the series we see that Haruka has a last name of "Shitow", not Mishima. Well, Mishima was her original last name, before her mother got re-married. The reason why Ixtli took the name of "Reika" instead of "Haruka" is because Ayato's memories of Haruka were essentially wiped out. But even Maya's powerful mind sweeping didn't erase all the memories of Haruka Mishima from Ayato's mind. Deep in Ayato's subconsicious the name "Mishima" remained, but the name "Haruka" didn't. Ixtli simply picked the name Reika out of the blue as far I know, not only did Ayato had only faint memories of Haruka (if any), it also needed to establish its own identity. That's why we see that Reika Mishima resembles a young Haruka in just about everything but hair color.