Obtaining a new trial requires more than proving that expert testimony relied on faulty science, however. “You have to show they wouldn’t have been convicted if jurors knew this information,” Ms. Garcia said. “That’s not easy.”

Ms. Bryan, a fourth-grade teacher, was found in her bedroom with four gunshot wounds on Oct. 15, 1985, in Clifton, Texas. She initially appeared to have been the victim of a burglary-turned-homicide. Mr. Bryan, a school principal, had been attending an education conference about 120 miles away in Austin at the time of the killing.

Sometime after Mr. Bryan returned from Austin, he let Ms. Bryan’s older brother, Charlie Blue, use his vehicle. Mr. Blue would later tell the authorities that when he opened the trunk, he found a flashlight with what appeared to be bloodstains on the lens.

It was among a series of details that led the authorities to focus more on Mr. Bryan; they arrested him eight days after his wife was found dead. And as lawyers squared off in the State of Texas v. Joe D. Bryan, the stains on the flashlight and in the bedroom appeared to help the prosecutors substantially.

They called Robert Thorman, a police detective who was trained in bloodstain-pattern analysis, as an expert witness. Pointing to the patterns of blood as evidence, Mr. Thorman tied the strands of the case together in a narrative that involved Mr. Bryan driving from Austin to Clifton, shooting his wife at close range, changing his clothes and driving back to the conference.

But in the years since, many key pieces of evidence, including the blood-splatter analysis methods used by Mr. Thorman, have come under suspicion. The case was investigated extensively by Leon Smith, a longtime reporter and newspaper editor in Texas, and other journalists.