Are tongues a sign for non-Christians or a sign for Christians?

I had, for a long time, been confused with 1 Corinthians 14:22-23. It says

So then, tongues are a sign not for believers but for unbelievers. Prophecy, however, is not for unbelievers but for believers. So if the whole church comes together and all speak in tongues, and unbelievers or uninformed people enter, will they not say that you have lost your minds?

I had been confused because in the first sentence, Paul said tongues were a sign for unbelievers, then in the third sentence, he said that if unbelievers come and see you speaking in tongues, they'll think you've lost your minds.

So which is it?

Upon further reflection, I think what Paul is pointing out is this:

1. If an unbeliever who knows a language you don't know comes and hears people in your church speaking in their own language, when it is clear that the speakers never learned that language, and then they hear it interpreted by another person who doesn't know that language, and the interpretation is correct, they will realize a miracle has occurred, and be drawn towards Christ by this sign.
2. If an unbeliever comes in to your church and hears everyone babbling in tongues without interpretation, they will think you've all gone crazy.

I think this makes sense in light of the fact that Paul was just before this making the point that we should not have “mass babblings” where lots of people are speaking in tongues without interpretation.