Twitter insists you add an email to your account in order to enable 2FA. Nope. I'm not giving them my phone number or email.
The workaround is to use an email forwarding service. I've been using https://simplelogin.io/ which is dead simple to use, and the free tier gives you quite a lot to work with.
Boom. 2FA enabled, and without Twitter ever knowing my real email address.
@voltronic Thanks!