I have solved my problem like SaraM.
Hoever, there is a caveat, that needs an additional tool:
If you "sync" Outlook tasks via iCloud, this actually means that the iCloud control center adds an additional data file into the Outlook profile. This data files contains a task folder which directly syncs with the iCloud task database (actually: reminder data base). This works fine: tasks in Outlook's iCloud task folder, on my iPhone and on my iPad are all in sync (some Outlook specific features as some recurring patterns are not supported - but that's nothing now).
The caveat is, that (a) the creation of new taks in Outlook as well as (b) the task list displayed in Outlook's right hand column are based only on the tasks in the taks folder of Outlooks standard
data file (same is true for contacts and calenders). Hence, I needed a tool that synchronizes Outlooks iCloud task folder with Outlooks standard data file's task folder. I am using OLMixedFolders
for that. It's new, works quite well and the support is very responsive.