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Given that the 46 chromosomes split evenly during meiosis, if the even pairs of the male gametes match with an even pair of the female gametes, the offsprings in theory should be carrying chromosomes the other does not posses, and thus in theory share no genetic material, despite coming from the same parents. This follows the assumption that each chromosome is unique and share no genetic materials with the other chromosomes.

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Chromosomal abnormalities aside, every human offspring has 23 chromosomes from mom and 23 from dad.

If chromosomes were inherited as whole units (they are not), the odds of one sibling, fraternal twin or not, having exactly the opposite set of chromosomes would be 0.5^46 = 1.4x10^-14, or about 1 in 70 trillion. That is, the odds of selecting the other chromosome 1 from dad is 0.5, same for chromosome 2, etc, and the same for each chromosome from mom.

Because of crossing over, the odds are even lower.

"Possible" is not very meaningful at these probabilities.

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The other answer correctly notes the probability is vanishingly small, equivalent to successfully calling a coin flip many times in a row. I'll also note that due to pedigree collapse, even two "unrelated" members of a population share some ancestor if you go back far enough - both your mother and father carry some small amount of identical DNA inherited from the same individual. For this bit of DNA, it doesn't matter if the offspring inherits from the father or the mother, they're inheriting the exact same DNA which can be traced back to the same ancestor. Fraternal twins can't differ at these otherwise normally variable loci, as the mother and the father don't have distinct DNA to give.

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