Could such a device alter the relative phase of two separate beams of light? This optical device would take as input two beams of light with some arbitrary phase difference, and output the same two beams but with their phases changed to some fixed offset relative to each other.

With such a device an actual switch becomes easy: Make the device change the two inputs to be antiphase and combine them. If one of them is missing, the other passes through unphased (pun intended), but if both are present they cancel each other out; aka you have an xor switch!

Yes, a conventional 2-port MMI will generally give you a pi/2 phase shift of one port relative to another. If you combine the 2 ports again with another MMI, you have built a Mach-Zehnder interferometer with ideally pi phase difference (if you've perfectly matched the length of the 2 waveguides) between the top and bottom signal paths. If you insert a phase shifter in one or both arms you can control the light at the output of the MZ by varying the voltage applied to the phase shifter. You can then modulate the voltage and produce an AM or PM signal at the output of the MZM. This is currently how some commercially available photonic communication IC's send data over the network.

There are limitations on how good the extinction (cancellation) can be based on how well the losses are matched in the respective waveguides.

In this case of this paper, I imagine that the phase relationship will be much more complex and it will highly wavelength dependent.

First, waves still pass through each other when destructively interfering. They temporarily cancel out each others' amplitudes, but not each others' change-in-amplitude, so they continue to propagate.

Second, XOR is not a universal gate (even when combined with NOT). XOR can only sum things together (mod 2). For universal computation you need to be able to multiply (e.g. with an AND gate or a Toffoli gate).