Writers and Publishers must share the same PRO, otherwise your registration will throw an error in our system as the work would be rejected by all societies.

Writers and Publisher are members of societies. When you initiate a registration, you're registering ownership of a song into the member accounts, so that the member can be identified and paid. Therefore, when you're registering songs to ASCAP, ASCAP wants to know the ASCAP writer members and ASCAP publisher members on a song so that they can properly match member accounts to works and pay accordingly. 

Writers own copyrights and administration rights until they assign those rights to a third-party publisher. The publisher must be a member of the same PRO in order to register the song on behalf of the writer. A publisher can, therefore, join ALL societies in the world in order to register songs into any given society on behalf of their writers who may be members of different societies.
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For example, Sony/ATV Publishing is a member of ASCAP, BMI, and SESAC. When Sony/ATV registers songs for their ASCAP writers, they register with the Sony/ATV ASCAP membership. When Sony/ATV Publishing registers songs for their BMI writers, they register with the Song/ATV BMI membership. And the same for SESAC. Sony/ATV cannot register with a song for their ASCAP writer with Sony/ATV's BMI membership. It will get rejected by ASCAP, because their will not be an ASCAP publisher account attached to the registration.
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So, Writer PRO ad Publisher PRO must match. Publishers can join multiple PROs in order to register for writers from different PROs.

In the event that they don't and you need to register your work, omit the publisher from the song so that the registration will go through TuneRegistry. If you want to register a song with unmatched PROs, you'll have to attempt that via the web portal for each society. They may still reject the registration.

Sub-publishers are a bit different and do not have to share the same PRO as the writer.
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Sub-publishers represent the publishing rights of an original/main publisher in a specific territory. By extension, the sub-publisher is the publisher of record in its territory. 

For example, Sony/ATV may hire PopArabia to be its sub-publisher in the Middle East and North Africa. So, PopArabia will register the song including the writer (ASCAP), Sony/ATV (ASCAP), and then list PopArabia as the sub-publisher of Sony/ATV in Middle East and North Africa (SACEM). In this example, the PROs of the writer and the original/main publisher Sony/ATV still must match, but the sub-publisher of the Sony/ATV does not have to match because PopArabia, the sub-publisher, is only representing the song in a specific territory.