Tire chains are not illegal in Minnesota when needed for safety and traction to get through the snow.
According to MN law on tire chains being allowed.
It shall be permissible to use any of the following on highways: implements of husbandry with tires having protuberances which will not injure the highway, and tire chains of reasonable proportions upon any vehicle when required for safety because of snow, ice, or other conditions tending to cause a vehicle to skid.
MN Prohibits the use of studded tires by residents although non residents coming through the state of MN are usually permitted to use studded tires.
According to MN law on studded tires for non residents.
Occasional use of studded tires by nonresidents.
Notwithstanding the provisions of subdivision 1, a person, operating a motor vehicle properly licensed and registered in another state or province of a foreign country which authorizes the use of tires with metal studs or wire embedded tires on its highways, may operate the motor vehicle with tires having metal type studs or with tires having wire embedded therein on the highways of this state while occasionally within the state, and such use while occasionally within the state is not unlawful; provided that, the metal type studs shall not exceed 5/16 of an inch in diameter inclusive of the stud casing with an average protrusion beyond the tread surface of not more than 7/64 of an inch, and the number of studs in a tire shall not exceed two percent of the total net contact area.
Use of a vehicle in this state on more than 30 days in any consecutive six-month period is not occasionally. Except, a person whose regular place of employment is within the state or who is a student at an educational institution located within the state, shall not operate a vehicle, regardless of its place of registration, upon any highway within the state if such vehicle is equipped with tires which would be unauthorized were the vehicle registered in this state.
In MN or Minnesota Tires are not allowed to stick out of fenders at all.
However if your tires are only sticking out past the fenders only an inch or 2 then it's unlikely you'll get a ticket or be bother by it.
However for tires that stick out past fenders in MN or Minnesota they require a fender flare to be legal.
According to Minnesota Law Installing wheels that extend past the fenders require a fender flare.
They must be at least as wide as the tires and must have a clearance of not more than nine inches from the ground when the vehicle is empty.
Minnesota law also states: "Every passenger automobile shall have fenders, or other devices, that are designed to prevent, as far as practicable, water, dirt, or other material being thrown up and to the rear by the wheels of the vehicle."
So every vehicle must have fenders.
If you get pulled over without fenders or your tires sticking out too far from the fenders you could receive a ticket.