How do I disagree without turning the other person off

I have seen this situation in a zillion books but when faced with this situation I just dont know what to apply from what I have read.

For the last few days I have two teams that are battling it out over a problem. Each of the teams has their own way of doing it and both solve the problem and both have their pros and their cons. I belong to one of these teams. The deadline is drawing closer and closer and both the sides are getting desperate for their solution to get selected. And the best part is the two teams are seperated by 6000 miles in two different continents and their only medium of communication is telephones and emails.

We have been arguing so much over which is the best way to solve the problem that it is becoming evident that both sides are getting fed up of each other and both the sides are succumbing to the "whatever" route. The result of this is that we are losing focus on solving the problem and we are ending up with a "whatever" kind of a solution with no clear winner and with both the sides feeling down.

The question is "How do you salvage such a situation and get the best solution selected ?" This looks so awfully similar to the situation exercises that one gets when you attend a Managemet Training Program. But now this is real life for me...

Will let you'll know how it went....