Just like in any language, there are grammar rules that aren't always followed by native speakers. This is one of them. Many native speakers will use the more informal "there's" for both singular and plural, especially when speaking. You can also use it in speaking and it won't be noticeable; however, writing can be more formal and you should try to use the correct subject-verb agreement. Informal writing like text messages could also use the "there's" without issue. I usually recommend as a non-native speaker to try to follow the grammar rules as much as possible. Great question!

Using "there's" with a plural subject is becoming widespread. It's not something I ever use and I don't recommend it, but I see and hear it regularly-- even from natives.

Notwithstanding, don't write out "There is both."