I faced this problem a few years ago. In that case, I obtained a satisfactory approach along the lines of one of your suggestions. Specifically, I found that the eigenvectors changed relatively slowly with $t$. So I could associate the corresponding eigenvalue curves after an intersection with the right ones before an intersection by considering the similarity of the corresponding eigenvectors. I found that the inner-product $X^T_i Y_j$, where $X_i$ is one of the eigenvectors after the intersection and $Y_j$ is one of the eigenvectors after the intersection was a good measure of their similarity.

By the way, you might get more answers by posting this on http://scicomp.stackexchange.com/

I faced this problem a few years ago. In that case, I obtained a satisfactory approach along the lines of one of your suggestions. Specifically, I found that the eigenvectors changed relatively slowly with $t$. So I could associate the corresponding eigenvalue curves after an intersection with the right ones before an intersection by considering the similarity of the corresponding eigenvectors. I found that the inner-product $X^T_i Y_j$, where $X_i$ is one of the eigenvectors after the intersection and $Y_j$ is one of the eigenvectors after the intersection was a good measure of their similarity.

By the way, you might get more answers by posting this on https://scicomp.stackexchange.com/