English:
or ・ possibly・maybe・perhaps・or possibly
Structure:
A + (か) + あるいは + B
A。 あるいは + B
あるいは + Phrase + かもしれない
[If used with かもしれない means “possibly”,“perhaps” or “or possibly”.]
I might be wrong, but I thought あるいは was in place of か, since they both have a similar meaning.
The Structure box shows that you can use か＋あるいは.
(I’m not used to seeing that either)
You can find people using it on twitter “効率を落とすかあるいはミスを隠すようになる。” “放流に伴う安全確保のためか、あるいはヒグマ出没のためか”
“自らか、あるいは然るべき人との壁打ちで”
Not sure why but the comma after か feels better.
Oops, I should have looked at that first!
In this case I agree 
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It seems it can’t be used in most cases. あるいはthe option isn’t accepted most cases.
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Is it normal for あるいは to appear twice in all these sentences in the “About あるいは” section? In the audio examples just below it, it just appears once.
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No should be just one. I’m not sure who to ping for this tho. If you take it to the bug report thread it’s more likely to be seen
https://community.bunpro.jp/t/feedback-bug-reports/130/3861
Welcome to the community 
Hey @snowwater, and welcome to the community !
It is not normal for あるいは to appear twice, so we have just fixed these!
Thanks for letting us know about this error!
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I find one of the example sentences rather odd:
牛乳あるいはチーズが使われている料理は食べられません。
I can’t eat anything that uses milk, or another possibility is cheese.
What is being conveyed here exactly? That the person isn’t sure if they can eat cheese or not?
When I first read the translation I actually misinterpreted it and thought it meant that the person was saying that cheese could replace milk in the recipe.
The Japanese sentence means “I can’t eat anything made with milk or cheese”.
I’m not sure about the English one.
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Thank you. I think what confuses me is that the point in general as explained in the bunpro lesson seems to carry some element of uncertainty, but there’s isn’t any here, right?
So basically in this case it effectively means と?
It means A or B, like AかB.
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晴れあるいは曇りであれば、お祭りに行きましょう。
Am I right in thinking that this means:
If it’s sunny, or even if it’s cloudy, let’s go to the festival. (but it’s better if it’s sunny, and we make no promises about other cases such as rain).
The impression I’m getting from the English translation is somewhat different, which is throwing me off a little bit:
Whether it is sunny or cloudy, let’s go to the festival. (it doesn’t matter which, we go there absolutely in any case).
I’ve re-read the linked 類語例解辞典 snippet about あるいは, and I don’t 100% understand it, but it seems that it typically puts more emphasis on option A as opposed to B.