Learn Spanish (Latin America): Tampoco me gusta la sopa.

In Spanish, when a negative word like tampoco (neither/not either) comes before the verb, you drop the no. You only use a double negative if the no comes first.

If I wanted to put tampoco at the end of the sentence, how would I write it?

You would say No me gusta la sopa tampoco. Because tampoco is now after the verb, you must put a no directly in front of the verb to make the sentence negative. Both versions mean the exact same thing.

The English translation says "either", but I thought tampoco meant "neither"?

It means both! English grammar forces us to choose between "neither" or "not... either" depending on how the sentence is built. Spanish just uses tampoco anytime you are agreeing with a negative statement.

Just as a refresher, why is la sopa at the end of the sentence?

Because with verbs like gustar, the thing you like is actually the subject of the sentence doing the action. The sentence literally translates to "The soup does not please me either." You can put the subject at the beginning (La sopa tampoco me gusta), but putting it at the end is the most natural word order in Spanish.