The Tour of Monte Rosa travels through Switzerland and Italy.
If you are a member of one of the respective alpine clubs, you get discount on accommodation at their mountain huts. Unless you plan to stay exclusively in alpine club mountain huts, it is not usually worthwhile getting membership only for the hike. Here are the details on each alpine club membership:
SWISS ALPINE CLUB MEMBERSHIP
The Swiss Alpine Club has a membership program that provides discounts to their mountain huts. Membership is from 80 CHF and is open to non-Swiss citizens. For full details click here. The huts along the Tour of Monte Rosa that are owned by the Swiss Alpine Club include: Britanniahutte.
ITALIAN ALPINE CLUB MEMBERSHIP
The Italian Alpine Club has a membership program that provides discounts to their mountain huts. Membership is from 45€ and is open to non-Italian citizens. For full details click here. The huts along the Tour of Monte Rosa that are owned by the Italian Alpine Club include: Rifugio Teodulo, Rifugio Crespi-Calderini, Rifugio Pastore, Rifugio Oberto.