If a group of originally transient explorers went to Mars with all
the supplies they need for a surface stay of at least one year
(which is what most manned Mars missions envision for a duration
on Mars itself), plus all the other supplies needed for the several
years to and from that world (assuming we go there via conventional
method - no fancy antimatter or Heim drives), what if they decide
to stay?

It won't be like with Apollo, where they only had relatively limited
resources. I could see a resourceful crew growing enough crops
to survive and eventually expand from there. As for building
materials, I know of several plans for making Mars habitats out of
the regolith by turning them into bricks - as an expansion from
their original lander and likely smaller bases they began with.

And who could stop them? Do you think Earth authorities would
send a police force after them? If they were smart, they'd send
even more colonists who may be loyal to the Earth authorities to
establish their own beachhead to grab resources and territory
before the original "renegade" settlers declared all of Mars for
themselves.

I can see it sparking quite a revolution in getting humans out of
our blue and white cradle. The next phase of human exploration
beyond the Earth-Moon system will not be as neat and tidy and
ultimately dead-end as Apollo was. At least if we do it right.