new data only sim ROAMLINE

[inquisitor]

You can definitely buy a German Aldi SIM as a foreigner (anything different would violate the freedom of services within the EU) but you'll need to provide a German address as their activation system does not support foreign addresses. In contrast to Fonic/Lidl I haven't heard of Aldi sending out confirmation letters to the provided address, so I believe you can provide just some arbitrary existing address.

As of recharging prepaid-wiki.de says that Aldi SIM cards can be recharged by bank transfers to eplus (who actually run German AldiTalk like the Belgian AldiTalk service is run by KPN Belgium) like it works for eplus' prepaid service called "Free & Easy" (described here).
Following this information you should be able to recharge your German Aldi SIM card by transfering the desired amount to the following bank account:

payee: E-Plus Mobilfunk GmbH & Co.KG
IBAN: DE07300400000121790001
BIC: COBADEDDXXX
reference:

Recharges by bank transfer prolong validity of your SIM card as follows:
< € 0.10 = no prolongation
>= € 0.10 = 3 days
> € 1 = 1 month

So you can test this method by transferring a cent or so. Once your payment has been received and the amount has been accredited to your account you'll receive an SMS confirming the recharge.