I am afraid I do not consider that to be a valid marriage. It is all about the cultural context but in many Western countries marriage is defined by the state in order to allocate responsibilities and privileges accordingly. For example a Church wedding is followed by some sort of registration with the civil authority.
On the spiritual side this could very well be a marriage but it is highly doubtful that they could enforce it legally. Given the culture wars that are raging at the moment, the state is taking a very keen interest in the definition of marriage and its implications. Therefore it is inconceivable that tying a mangalsutra is going to be accepted as a marriage ceremony.
I would be very interested in the perspective of someone who has been through this process.