This is a known problem and it has been for the past four years: https://success.salesforce.com/ideaView?id=08730000000oMVgAAM
For myself, I am going to find out if I can replace the process with an old fashioned work flow. Update: this turned out not to be possible. The triggering record was a Task and the record that must be updated is an Account. Just the kind of thing that a process can do, but a work flow can't.
The solution for me was to do the update not immediately, but in the future. I moved the actions from direct to scheduled (well, moved: no, I copied them and then deleted the original). I used a one hour delay.
That way, the update is done way after the other update/creation has been done and there is no conflict. For me this works, because the concerning field does not need to be updated in real-time. A one hour delay is acceptable.