This is commonly known in Germany as "Bauleitung".
The only problem is that this word is used both by the construction companies who carry out the work and the architects themselves.
In England the architects don't supervise the work on site as you have described. This is carried out by a Clerk of Works who is responsible to the architect and the client.
Contracts Managers are responsible for the work on site for the construction company and they are the partner for the architect.