You definitely have to have matching voltages as Dogman stated, so for ease of swapping things around you should run the wife's pack as dual 36's.
It's best not to mix different chemistries in parallel, because they have different discharge curves. To get a decent match of voltage with Lipo and your 24s Lifepo, you'd need something like a 21s pack of Lipo, which you couldn't split in half for the wife's ride. With the differing discharge curves what would happen is the Lipo would supply the bulk of the power early the ride, then the flatter curve of the Lifepo would take over until late in the ride where the Lipo would take over again.
Another route would be to put your dual 36's in parallel, and take the wife's pack and put it in series with your 12s4p of headways. As long as your controller can handle the new voltage, that could work, but you'd want a close match in capacity. You'd also need to monitor the voltage of both individually and avoid running one dry. I don't like this route.
The best would be dual 36's of headways for the wife too, but that ain't cheap. Personally I'd go economical and get the Lipos for me and set it up with a permanent 21s. Then assuming a 3.65V/cell top of charge for the Headways, for the long distance rides I'd charge my Lipos to 4.17V/cell, and put the two 36V Headway packs in series with each other and parallel that with the permanent Lipo pack on my bike. The Lipos and the Headways are individually sufficient to carry the load, so it's wouldn't harm either one. Plus the top of charge voltage and low voltage of the 2 packs are close enough that the whole thing works. The 21s of Lipo is a bit of an oddball for charging, so you'd end up having to invest in 24s charging/balancing capability for the Lipo.
Another route would be to never connect the 2 packs together. Make your run out using the lower capacity pack, and when that one is used up, turn around. Disconnect the depleted pack, and connect the full pack for the ride home. What I don't like about this approach is fully discharging a pack, though you could just make your halfway point 80% DOD of the smaller pack. The good part is that you don't have to worry about matching voltages, and you have more juice to get home than you used going out, so you're covered for a little headwind or a faster trip home.
John