Both spouses are free to believe as their hearts feel is right. However it would be advisable to discuss this with each other freely and at length. The most important question would be if, in your example, the wife insists on a path that involves "shirk" or idolatry, or if the belief is something contrary such as believing in Ali as a prophet or believing certain of sahhaba (companions) or wives of the prophet are going to hell. Such a belief is unacceptable and would invalidate the marriage should the wife become an idolator (mushrika) or otherwise reject Islam and that Mohammad is the seal of the prophets. But discussion and patience should precede such a step.
A few important issues were not discussed in the above answer which should be addressed. First, you mentioned that this example (Sunni wife becoming Shi'a) is not common, and requires certain uncommon circumstances. In this example, unlike joining another entirely different religion/ belief system, such as Taoism, both sects agree on certain basic things: Allah is one, Mohammad is His messenger/ prophet, Resurrection and Judgment after this life, the angels, and the Quran. Within each sect are variations as to practice (schools of thought) and details regarding belief, such as whether or not Prophet Mohammad was m'asoum or free from mistakes. Then there is, further, a certain degree of leeway where Muslims may agree to disagree, such as whether or not a woman can pray wearing clothes made with printed cloth or the whole issue of wearing black, white, or colors. Disagreement over such minor issues should not cause a rift in a marriage; even outside the subject of religion there will always be such trivial disputes.
During the time of Prophet Mohammad, Allah ordered to change the qibla which Muslims face in salat. This was a test of faith; would the believers follow Mohammad's directive, knowing it was from Allah, or would they dispute this break from tradition? In Surat al-Baqara 2:143, we read:
And thus we have made you a just community that you will be witnesses
over the people and the Messenger will be a witness over you. And We
did not make the qiblah which you used to face except that We might
make evident who would follow the Messenger from who would turn back
on his heels. And indeed, it is difficult except for those whom Allah
has guided. And never would Allah have caused you to lose your faith.
Indeed Allah is, to the people, Kind and Merciful.
The lesson here is not only that we must obey the prophet even when he orders a change that breaks with a major established norm, but also that Allah's word supersedes tradition or "established norms". There is a sense in which the Shi'a/ Sunni schism is more about history, traditions, and tafseer or interpretation than about the bedrock of faith; for that reason there is an established norm of both sects joining in the one Hajj, an established norm I applaud for its acknowledgement that faith supersedes these differences, or can if we don't get bogged down in details.
Taking this further, there is a scenario in which the wife could be Shi'a and the husband Sunni and it could work; but only if both agree on the "high road" on both sides, that is, with their strongest mutual commitment to the bedrock of common faith, and willingness to compromise if one can show the other that a particular detail contradicts that bedrock of faith. This is far more likely to work if they married like this; for her to suddenly convert unexpectedly to Shi'a, there would have to be discussion as to why, what attracted her to the change and why or if it is important to her. If it is a superficial matter at heart, such as her having gone to a Shi'a mosque and gotten along better with the Muslimat there, maybe. But as you said, it would be difficult, especially if her husband saw this as kufr.
The issue is so fraught with details and subjectivity. But if the dispute brought irreconcilable differences, it could end in divorce. Then what if she changed her mind again — changes of this nature could be a sign of a vacillating state of mind and lack of commitment — would he take her back? And if she is undecided at heart, why break off from one and jump into the other without giving it time?
There is also a danger for anyone who seeks to change from Islam over some details that are made clear in the Quran:
In Surat Al-Shura 42:16;
And those who argue concerning Allah after He has been responded to -
their argument is invalid with their Lord, and upon them is [His]
wrath, and for them is a severe punishment.
If the spouse who changes their sect or religion,m had read and understood something about their faith to begin with and then after marriage wanted to change, that change should not be entered into without seriously thinking about it. More than the marriage itself, the wrath of Allah, were their decision to turn away from Him, would be a far worse outcome than divorce. And that issue may be in the mind of the unchanging spouse, but should be on the mind of both.
