You could think of integers as pairs of natural numbers $(a,b)$ modulo the equivalence
$$(a,b) \sim (c,d)\quad \mathrm{if}\quad  a+d = b+c.$$
(In other words, you think of $(a,b)$ secretely as $a-b$.)  Addition is just entrywise addition of natural numbers:
$$(a,b)+(c,d) = (a+c,b+d).$$
Negative numbers are numbers of the form $(0,a)$.  Now the equation $-7 -3 = -10$ is
$$(0,7) + (0,3) = (0,10).$$
In summary, you can restrict yourself to adding natural numbers, provided that you consider pairs.

I'm not sure this helps in your friend's case, though.

**Edit**

Negative numbers do not arise in nature, of course.  It is the result of comparison.  The above set up with pairs is precisely that.  $(a,b)$ is positive if $a>b$, negative if $a<b$ and zero if $a=b$.  This could perhaps be woven into some sort of narrative for your friend.