Let $(a,b,c,d)$ be a quadruple of positive integers, and let us want to check if $ab=cd$, $ab>cd$ or $ab<cd$. The case when some of the numbers $a,b$ equals some of the numbers $c,d$ is trivial, so let us assume that $a\ne c$, $a\ne d$, $b\ne c$, $b\ne d$. If $a>c$ and $b>d$ then the situation is also trivial, and so is it in the case when $a<c$ and $b<d$. If $a>c$, but $b<d$, then $(a-c,b,c,d-b)$ is a quadruple of positive integers whose sum is less than $a+b+c+d$, and the equality $$(a-c)b-c(d-b)=ab-cd$$ holds. If $a<c$, but $b>d$, then $(a,b-d,c-a,d)$ is a quadruple of positive integers whose sum is less than $a+b+c+d$, and the equality $$a(b-d)-(c-a)d=ab-cd$$ holds. This obviously shows a solution of the problem by an algorithm using only comparing of positive integers and subtracting them in the case of positive difference. The algorithm can be somewhat improved by using that the following two cases are also trivial: the one of $a>d$, $b>c$, and the one of $a<d$, $b<c$ (another reduction similar to the above one can be performed if none of these two cases is present).
Example. Having in mind the initial question in this thread, let us apply the above-mentioned algorithm to the quadruple $(254,847,383,536)$. We get consecutively the quadruples $(254,311,129,536)$, $(125,311,129,225)$, $(125,86,4,225)$, $(121,86,4,139)$, $(117,86,4,53)$, and since $117>4$ and $86>53$, it follows that the product of $254$ and $847$ is greater than the product of $383$ and $536$.
Remark. One can get the result in the above example by means of several shorter sequences of quadruples if sometimes the other reduction possibility is used instead of the one which was described above. Here are some of these sequences:
$(254,311,129,536)$, $(125,311,129,225)$, $(125,86,4,225)$, $(125,82,4,100)$;
$(254,311,129,536)$, $(125,311,129,225)$, $(125,182,129,100)$;
$(254,311,129,536)$, $(254,182,129,282)$, $(125,182,129,100)$;
$(254,464,383,282)$, $(254,182,129,282)$, $(254,53,129,28)$.
The first of them would be actually produced by the transformations done in Arjang's answer from Dec 16'10 after the correction of an error made there (it must be 311 instead of 310).
Problem. Is it possible to design an algorithm using the same primitive operations which compares $x_1x_2...x_n$ and $y_1y_2...y_n$, whenever $n,x_1,x_2,...,x_n,y_1,y_2,...,y_n$ are positive integers?