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On the one hand, I don't immediately see how a correspondent or coherent theory of truth would directly prohibit the truth of two contradictory statements at once – but this is not an argument, I may simply not know them deeply enough. At least this is one of the objections to the coherent theory – the hypothetical possibility of two or more internally consistent systems that consider different statements about the whiteness of snow to be true. However, the counter-argument is the assumption that there can be only one perfectly complete system, that is, although a coherent theory does not prohibit contradictory true statements, it at least hopes that they are impossible. And in any case, their truth will exist only within the system, that is, for each other, at the same time, they will not be true.
On the other hand, Popper has a famous proof that if the law of the excluded third is eliminated from formal logic, allowing the simultaneous truth of opposite statements, the very meaning of “truth” as a condition inherent in the statement is lost, everything becomes true in general. This can be considered an indirect proof that it is impossible to consider opposite statements as true at the same time, without shaking up the entire logical apparatus in general. That is ,the “truth” in a system that allows the truth of two opposite statements will be something completely different from the usual truth.