The Crown made the request Tuesday at the trial of Michael Marin, 63, who pleaded not guilty to six charges stemming from allegations that date back to 2004 and continue though 2015.
A court order prohibits reporting any information that may identify the complainant.
Prosecutor David Kirk told the court the woman, now in her 20s, has difficulty talking about the victimization that is alleged to have begun when she was 10 years old.
There is evidence that she has been diagnosed with post traumatic stress disorder as a result of sexual abuse, the assistant Crown attorney said.
"She also suffers anxiety as a result of these incidents," Kirk said, calling them "intrusive sexual assaults" that occurred over a long period of time.
"She still has heightened anxiety being in the room with him."
Kirk noted that a news reporter was in the courtroom, which also might cause her increased worry about testifying.
The woman is going to be talking about deeply personal things, which is hard to do when put in a "room of strangers," he said.
Defence lawyer Bruce Willson opposed what he called the Crown's "extraordinary" application.
"An accused has the right to face their accuser."
He told the judge he has a problem not having the woman in the courtroom and not being able to see her face.
Willson, however, said he had no issue with the woman testifying behind a screen in the courtroom.
"All the screen does is prevent her from seeing him. It doesn't prevent you or me from seeing her."
Gareau said he was going to think about the application over night, but indicated two things were troubling him.
The judge said there was no medical opinion on how testifying in open court would impact the woman and there was no direct evidence from the complainant.
"This is a close call as far as I'm concerned."
The Crown called one witness, a city police officer involved in the investigation, to testify at Tuesday's hearing on the application.
Det. Const. Alison Campbell said the "very shy" woman has a hard time speaking in front of people, and believes she will not be able to tell all the details she wants to sitting in front of the accused.
"This morning she said she's on egg shells, has a fear of running into him."
During cross-examination by Willson, Campbell said that the complainant doesn't want to be in the same room as Marin.
She agreed with the defence that no one, but the woman herself, had indicated that she's not able to testify and that she has an anxiety disorder and PTSD as a result of sexual assault.
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