The short answer is that the court will rely on non-genetic evidence, such as the testimony of the mother.
In 2007, the Missouri Court of Appeals ruled on a case in which twin brothers Richard and Raymon Miller both engaged in sexual intercourse with Marie Adams, who gave birth to a child the following year. Both the State and Ms. Adams sought a court order establishing which of the Miller brothers was the father, presumably in an effort to obtain a child support order in favor of Ms. Adams.
Identical twins share identical DNA, so blood tests demonstrated that both brothers had a 99.999% probability of being the father. This “hard” evidence excluded everyone in the world other than the two brothers, but could not establish which of the Miller brothers was the father.
Next, the court considered “soft” evidence, specifically the testimony of both men and the mother. Both men admitted to having sex with Ms. Adams on or near the date of conception in August of 2003. Ms. Adams testified that she did in fact have sex with both brothers in August of 2003. However, she also testified that when she and Richard had sex on August 22 or 23, he used a condom; whereas, when she and Raymon had sex on August 8, he did not use a condom.
Based entirely on this testimony, the trial court ordered that Raymon was the father of the minor child.
For more information, see In State ex rel. Department of Soc. Servs. v. Miller, 218 S.W.3d 2 (Mo. App. S.D. 2007)