They have a right to be present but can be removed under Illinois v. Allen
While in general, a person has a right to be present for the trial and conviction hearing, they can be removed if they disturb the orderly process. SCOTUS also offers to have him fined again and again (only works if the accused is not judgment proof) or to gag and bind an unruly defendant.
Illinois v. Allen, 397 U.S. 337 (1970)
Respondent, who was on trial for robbery, was removed from the courtroom for repeated disruptive behavior and the use of vile and abusive language directed at the trial judge, notwithstanding the judge's prior warning that removal would follow another outburst. Appointed counsel represented respondent during the period respondent was not allowed in the courtroom, principally the presentation of the State's case. Having given some assurances of good conduct, respondent was allowed to return to the courtroom while appointed counsel presented his defense. Respondent was convicted. Following the State Supreme Court's affirmance, respondent filed a petition for a writ of habeas corpus in federal court, contending that he had been deprived of his right under the Sixth and Fourteenth Amendments to confront the witnesses against him. The District Court declined to issue the writ. The Court of Appeals reversed, holding that a defendant's Sixth Amendment right to attend his own trial was so "absolute" that, regardless of how unruly his conduct, he could never be held to have lost that right so long as he insisted on it, as respondent had.
A defendant can lose his right to be present at trial if, following the judge's warning that he will be removed if his disruptive behavior continues, he nevertheless insists on conducting himself in such a disruptive manner that his trial cannot proceed if he remains in the courtroom. He can reclaim the right to be present as soon as he is willing to comport himself with decorum and respect. Pp. 397 U. S. 342-343.
A trial judge confronted by a defendant's disruptive conduct can exercise discretion to meet the circumstances of the case, and though no single formula is best for all situations, there are at least three constitutionally permissible approaches for the court's handling of an obstreperous defendant: (1) bind and gag him as a last resort, thereby keeping him present; (2) cite him for criminal or civil contempt; or (3) remove him from the courtroom, while the trial continues, until he promises to conduct himself properly. Pp. 397 U. S. 343-346.
On the facts of this case, the trial judge did not abuse his discretion, respondent, through his disruptive behavior, having lost his right to confrontation under the Sixth and Fourteenth Amendments. Pp. 397 U. S. 345-347.
Example of removal in action: The Darrell Brooks case
As a rather public example, this happened during the Darrell Brooks trial, multiple times: there's abouht 9:12 hours and 6:15 hours of sentencing hearing in this case, including breaks, split between two days. On the 2nd day, the judge starts the actual sentencing by elaborating on how she understands the case at about 3:11:45. If you look up the transcript, keep in mind that the local time at the start of the video was about 12:50, ten minutes to 1 PM.
At 3:44:20, Mr. Brooks starts to speak up and interrupt the judge, receiving in quick succession multiple warnings that he will be removed if he doesn't stop. After 40 seconds, the order to remove him from the courtroom is given.
The hearing resumes at around the 3:49:50 mark, Mr. Brooks now in a separate courtroom. The judge elaborates that Mr. Brooks has a right to be present, and also why he can be removed under Illinois v. Allen. There is more elaboration on this topic till Mr. Brooks is muted at about the timestamp 3:53:00, because he is unwilling to do the required pledge to reclaim the right to be present under that case. More explanation about it follows that the judge wants the pledge in writing at that point.
The reading of sentencing factors only resumes at around the 3:55:00 mark. Mr. Brooks writes something at about 3:57:00, and the paper is given to the judge at 3:59:19, ruling that whatever is on the sheet is not a pledge and thus he stays removed at 3:59:25.
For some hour Mr. Brooks waves papers at the camera, writing another note to the Judge at starting about 4:41:00, before resuming waving with the new note at about 4:43:20 and showing his statement to the camera at 4:55:52. Sadly the quality is not good enough to read the letters, and not even the judge can read it at 5:57:00 - re-iterating that unless it is a pledge to stay quiet, he's not coming back. The prosecution manages to make out the top word as Objection at 4:57:20 and shortly after it is ordered that he may return with the sign, but if he so much as speaks up, he will forfeit his right to be present.
Court resumes at about 5:03:20, corresponding to 4:53 in the courtroom. Mr. Brooks is instructed that he can only stay in the courtroom if he is to waive his right "to do so", and that if he doesn't behave he'd be removed again and told to sit down. Instead, Mr. Brooks starts rambling about subject matter jurisdiction at 3:04:00. Told once more to sit down, this transpires:
Mr. Brooks: "I'd like to go back to the other court room."
Judge: "It's not a courtesy to you. If you like to specifically waive your right to be physically present, then I will entertain that. Otherwise, you need to sit down."
Mr. Brooks: [something unintelligible] "I never waived a right to not be present" [more unintelligible]
Judge: "That's because you forfeited your right to be present by conduct. You're now back in this courtroom.
Mr. Brooks: "Your honor I wrote three... I did what you asked me to do..."
Judge: "You never once pledged to me sir, that you would not interrupt, and you're demonstrating by being here that you will continue to interrupt.
Mr. Brooks [something to the effect that the judge told the bailiff that the pledge had to be in writing]
Judge: "Mr. Brooks, this is not a debate. You asked to come over here and I honored that, and I brought you back.
Mr Brooks [Interrupts, the words My Right are understandable.] "three times. I shouldn't have to do it three times!"
Judge: "None of the opportunities you wrote to me said *I pledge to not interrupt."
Mr. Brooks [Interrupts] "I never had to do that before! You've never required that before!"
Judge: That is actually not true sir.
Mr Brooks [argues that he'd never been required to pledge like that before in trial]
Judge: Mr Brooks, You were just simply trying to delay the inevitable, please sit down.
Mr. Brooks continues to argue and stand up, despite now being told to sit down and be quiet for the 3rd time since returning to the courtroom. A 4th order to "please sit down" is issued at about 5:05:35 and two more within about 20 seconds later. Mr. Brooks rants and complains about being removed in the first place and that he'd want to exercise his right to be present. The Judge finally re-iterates at 5:03:40 that the reason he was brought back was not because he had complied with the request to pledge not to interrupt, but because the next portion is very important. Mr Brooks doesn't sit down and argue. At 5:08:00 the judge clarifies that he "did not reclaim his right to be present" (which was forfeited back at timestamp 3:44:20!). A minute later, the judge orders him directly - with a warning that this is a direct court order and not doing what is told is contempt of court:
Judge: Sit down and be quiet, so I can make the appropriate record.
Mr Brooks [dispnbeys]
Judge: All right, he's not going to obey, he has now forfeited his right to be present He will go into the other court room. We are in recess until he's there.
5:09:25 -The camera pans to the clock showing 4:59:20
Around timestamp 5:13:20 trial resumes, and the judge once more reiterates that and how the court deems Mr. Brooks having forfeited the right to be present for about two minutes. His sign that he objects to the other room and wants to return is ignored, even as he parades it around. Sentencing continues without him. The sentences are read starting timestamp 5:55:20 - no further interruption happens.