Sure you can, you just need an anonymous voting mechanism that's sufficiently naive. You use the verifiable process to restrict access to that anonymous mechanism.
In Canada, at both federal and provincial levels, you walk up to a desk and identify yourself, are crossed off a list, and handed a paper ballot. You go behind a screen, mark an X on the ballot, fold it up, take it back out to another desk, and put it in the box. It's extraordinarily simple.
> At some point, you have to give someone access to a database and they can change that database.
Well, that kind of fraud is a different issue from someone reading the database and figuring out who someone voted for (you just... don't record identities in the database).
In Canada, at both federal and provincial levels, you walk up to a desk and identify yourself, are crossed off a list, and handed a paper ballot. You go behind a screen, mark an X on the ballot, fold it up, take it back out to another desk, and put it in the box. It's extraordinarily simple.
> At some point, you have to give someone access to a database and they can change that database.
Well, that kind of fraud is a different issue from someone reading the database and figuring out who someone voted for (you just... don't record identities in the database).