There are no laws the prohibit this, but many employers have specific policies that do prohibit this. More common, though, are policies that prevent one family member from supervising another.
In this case, because the two are not legally married, they might look at other policies and see if the employer generally regards domestic partners the same as married couples. For example, if benefits and other privileges are the same, the employer is (most likely) treating domestic partners as if they were married. But if the employer does not, then your brother and his wife may be able to appeal to human resources by showing that the company does not automatically define domestic partners as "married" and this is one other instance where the company should not define define them the same.
It is also possible to appeal for an exception to the rule, or a transfer for one employee to another department, based on the circumstances.
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Disclaimer: The materials provided below are informational and should not be relied upon as legal advice.
I am not clear based on your description, what conflict the employer is pointing to in asking the couple not to work the same shift. I am not aware of any laws prohibiting a husband and wife to work the same shift. However, most companies have internal policies that prevent employees from dating each other, etc. Ask the manager for clarification, and consult your own attorney to protect your legal rights.