Reverse Charge (Steuerschuldnerschaft des Leistungsempfängers)

You invoice without German VAT and add a reverse charge note and both VAT IDs. Your client accounts for the VAT in their own country.

It also works the other way: if you buy services from abroad, for example software subscriptions, you may owe German VAT on them yourself.

General information on German tax law, not advice on your individual case.