Many people only know a franchise by its English name, but some don't even have an official English name / title. What should be our policy on that?
I'd personally prefer the romanized Japanese and a tag synonym from the English one.
For starters, we shouldn't just let anyone use any language they want for the tag names. We need to be consistent. Having some series tags in Japanese, and others in English is just confusing. We should strive to make things less confusing, not more. I think we can all agree that being consistent with language use is an easy way to make things less confusing.
Second, I think we can all agree that we should have a primary tag (to be consistent), but also have a synonym in the other language so that users searching for either language can find their questions. Remember, this is all about making things less confusing for the users.
Finally, the language we should use for the primary series tag should be the English name of the series, and the Japanese name should be a synonym. There are two reasons for this, the first being that the official language of Stack Exchange is English. I don't see any reason to use the Japanese name when we're all going to be typing in English.
The second, more important, reason, is to make things easier for users. Although I'm sure a sizable portion of our beta user base will be familiar with both the Japanese and English names of the series, this isn't going to be true for the community at large. After all, when you first starting watching anime did you have any idea what the Japanese name for any of the series you watched was? Probably not (assuming you're not from Japan). With this in mind, it makes sense that your average user will be searching for the English name, and not the Japanese name. So going with English just makes more sense. Remember, we want to make things less confusing for users, not more.
EDIT: In situations where the series doesn't have an English name going with the Japanese one is the obvious (And only) solution.
Here's my idea:
All types should be allowed. But we may need to pick a primary and synonymize all the others to it.
As for which we pick, I think that will depend on which one is most well recognized. (which may be different for each series) So I suggest we let multiple tags propagate first. And after a while we synonymize all the less used ones to the most used one.
This will keep things organized and together. So we don't get the questions split across 3 different versions of the same tag. When someone enters any one of the tags, it'll auto-correct to the primary.
It may be difficult to search for if the primary tag does not match what people are searching for.
So suppose we synonymize: senjou-no-valkyria to valkyria-chronicles All questions about that Anime will be tagged valkyria-chronicles, but searching for "Senjou no Valkyria" may not yield the question.