[quote=Peik]
How did Persia become Shi'a (or rather, what happened to the Safavid tariqa)? How did the Ottomans convert?Oh, and, I'm interested, obviously.
[/quote]

Illkhanate.

Before the Ottomans become the Ottomans they were historically a subject of the Illkhanate. The Mongol Illkhanate favored and converted to Shi'a while they ruled over Persia. Under this presumed time-line one of Osman's heirs could have presumably followed suit by either insistence by the Khan or some other factor. When the Mongols withdrew, the Ottomans were left behind and did as they did. Not as Sunni, but as Shia.

The means are not wholly important to the scope of the RP which for the large part is centered on the Mu'tazilah inspired Sunnis.