Consider a Stern-Gerlach apparatus that measures the $z$-component of the spin of an electron. Suppose our electron's initial state is an equal superposition of $$|\text{spin up}, \text{going right} \rangle, \quad |\text{spin down}, \text{going right} \rangle.$$ After going through the apparatus, the electron is deflected according to its spin, so we get $$|\text{spin up}, \text{going up-right} \rangle, \quad |\text{spin down}, \text{going down-right} \rangle.$$ In a first quantum mechanics course, we say that the spin has been measured. After all, if you trace out the momentum degree of freedom, we no longer have a spin superposition. In simpler words, you can figure out the spin by which way the electron is going.

In a second course, sometimes you hear that this isn't really a measurement. In fact, it is possible to "recombine" the beams back into $$|\text{spin up}, \text{going right} \rangle, \quad |\text{spin down}, \text{going right} \rangle.$$ at which point the original spin superposition is restored, just as coherent as before. For example, you could do this by putting the electron through a second Stern-Gerlach apparatus oriented upside-down relative to the first. This point of view is advanced in this lecture and the Feynman lectures.

Now here's my problem with this argument. Why doesn't the interaction change the state of the Stern-Gerlach apparatus? I thought the two states would be $$|\text{spin up}, \text{going up-right}, \text{SG app. down} \rangle, \quad |\text{spin down}, \text{going down-right}, \text{SG app. up} \rangle.$$ That is, if the appratus pushes the electrons up, it itself must be pushed down by momentum conservation. After "recombining the beams", the final states are $$|\text{spin up}, \text{going right}, \text{SG app. down} \rangle, \quad |\text{spin down}, \text{going right}, \text{SG app. up} \rangle.$$ and the spins cannot interfere, because the Stern-Gerlach part of the state is different!

This issue is a special case of a larger question I have: under what circumstances can interaction with a macroscopic system not cause decoherence? Intuitively, there is always a backreaction from the spin onto the system, which changes its state and destroys coherence.

In the case of a magnetic field acting on a spin, like in an NMR apparatus, there is a resolution. The resolution is that the system state is a coherent state, because it's a macroscopic magnetic field; and coherent states are barely changed by $a$ or $a^\dagger$. However, I see no such analogous resolution for the Stern-Gerlach apparatus.