It occurs to me that you can't show that two languages are related
simply by showing consistent diachronic changes. For example consider
language A with a consonants and x vowels, and language B with b
consonants and y vowels. Then regular sound correspondences can
always (or almost always) be found starting from language Proto-A-B
with (a+1)(b+1)-1 consonants and (x+1)(y+1)-1 vowels. To see how this
works consider the two made up words zig and bog. To account for the
b-z correspondance we assume the proto-language had a voiced
fricative bz. The sound change laws would be bz> b and bz > z. The i-
o correspondance indicates some close-mid central vowel. The g goes
unchanged. If in another set of words b corresponds to k you find a
sound which is somehow intermediate (maybe some type of labiovelar).
The formulas above take into account zero forms for those cases where
a consonant in one language corresponds to a vowel in another
(obviously the consonant disappears). Because of the large number of
possible proto-words most won't even be used. That would leave gaps
in the phonology and phonemes could be moved around to make a more
natural looking system. Likewise some sounds would be so rare that
conditioning environments could no doubt be invented to explain how
they could arise from sounds that are more common. Theoretically the
same could be extended to even more language groups (e.g. 4 languages
could be split into two groups of two, and then these two groups
related). With a sufficiently small set of cognates you wouldn't even
need to postulate huge phonological systems. By the same methods we
could relate affixes to each other.

So my question is, how could we know the difference between real
relationships and fake ones?