[J. R. Feehan, Anthony Padilla]
All Rand can remember of her was her smile and her hands. Nynaeve said it was obvious that she loved Rand, and that she was very nice. Even still, she was only in the Two Rivers for a few years, if all Rand can remember is her smile. She probably had known Tam for quite a while, while he was in Andor. At any rate, their relationship went on long enough for him to say in his fever dream that she "Always said you wanted to have children." That "always" would imply that her and Tam didn't get hitched after like a month-long romance, and that maybe they'd been married a while before they found Rand, and maybe tried to have kids of their own, to no avail, which would have taken a while to find out.
At a book signing, RJ is reported to have said that we'll find out a little more about Kari later.
Loony Kari theories: People have thought that Kari was Tigraine (somehow still alive after dying on the slopes of Dragonmount) or an Aiel Wise One looking after Shaiel's kid (which cannot be because Moiraine said that Kari's from Caemlyn) or an Aes Sedai who'd been stilled (someone said she was wasting away and that's why she died and Tam was her Warder, etc. Wouldn't the White Tower have gotten to Emond's Field faster if Kari had been one?) or a Tinker or she was related to some Caemlyn noble we know. Don't know why exactly, but she does have red hair, and the Caemlyn nobles also have red hair.