In the standard orthonormalization procedures, the first vector is important since it will define the first vector that will define the rest of the orthonormalization process.
The frame should be the same if a) you have the first two orthonormal vectors be the same (as calculated by the Gram-Schmidt and the first vector will always be just the first one in your list that is normalized) and b) the orientation of the set of both vectors is also the same (which means you need to consider the determinant of the matrix of all vectors to see if it's positive or negative).
If the above is the case, then you should always generate the exact same orthonormal basis for both sets.
The reason for the orientation is that basically different directions change the chirality: for example i X j = k by i x -j = -k.
Basically the orientation property will take care of most of it, but ultimately to get the exact same frame you want the first two orthonormal vectors generated to be exactly the same and if the orientation aspect is correct then the rest will take care of itself.