I recently met my cousins, but the question was brought up between us about the specific name.

We aren’t related by blood.

My father had a sister. His sister married a man who is now uncle. My uncle remarried and had two kids with his new wife.

closed as off-topic by Roger Sinasohn, ab2, Cascabel, AmE speaker, AndyT Jul 13 '18 at 8:19

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    You're not cousins at all. If your uncle by marriage divorced your father's sister and remarried and had two kids, I suppose if you were to make up a name for your relationship to them, you might call them your half-first-cousin-in-laws by divorce. – Billy Jul 13 '18 at 0:16
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    The problem with making up a name as @Billy did (and I like what he made up!) is that you will always have to explain it. If you feel cousinly towards them, and they feel cousinly towards you, then call each other cousins. If you don't, then just say you are sort of connected by marriage, but not by blood. English has no specific words for many family relationships. – ab2 Jul 13 '18 at 0:31
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    I'm voting to close this question as off-topic because there is no answer to the question. English does not have a word or a phrase for this -- and many other -- complicated family connections. – ab2 Jul 13 '18 at 0:33

All of humanity is related, with the cousin relation specified by degree and removal in English. That is, in English, the 'type' of cousin you are, as you put it, is something like "first cousins, twice removed", with the appropriate numbers used in place of first and twice.

The removal of the cousin relationship is the number of generations the cousins are apart. ... The degree of the cousin relationship is the number of generations prior to the parents before a most recent common ancestor is found. - wikipedia

All this, though, needs to be traced through ancestry, not marriage per se. So to find out what kind of cousins you are, you'll need to first find a common ancestor, which you haven't specified. If, for example, that ancestor was your great, great grandfather, and that you and your cousins are of the same generation, then you would be second cousins (count the number of 'greats' between you and the common ancestor, then count the number of 'greats' between your cousin and that ancestor, then pick the smaller number). If your cousins have one more 'great' to their ancestry (i.e. one generation your junior), then you'd be second cousins once removed.

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