Currently I have an option to use the "External team" feature, but unfortunately it requires some real email addresses. I just want to keep track of tasks I assigned to some members.

Hi @glebsam, I am afraid it's not in our consideration to add virtual team members. One thing that might help is to create an empty External Team (without inviting any actual members) as a workaround. This way, you can keep track of the assigned tasks by assigning to the External Team.
Hope this helps. Sorry again I don't have what you looking for, but please let us know if we can answer anything else for you. Thank you.
Golda, May 25, 2022
I thought about a fake email address, but it's not a right way to do it IMO. It's a crutch. Moreover, as you said, I'd have to pay for it (currently I am on Professional plan). So, for me virtual assignee would be a way to go.
G, May 24, 2022
Hi @glebsam, Please note that External Team members will be counted to the total member seats per organization. For example, if your organization has two member seats, you can invite one of them as an External Team member.
If you would like to invite virtual members, we allow inviting using a fake email address. For example, the actual email address is abc@gmail.com. Instead of using the real email address, you can input it as abc@email.com, and it will be shown ABC in the assignee icon.
You can try and let me know if you have any problems.
Golda, May 23, 2022