Where did Bellatrix Lestrange live after she left her husband? It never really says she "left" him so I'm assuming she did since she was so totally smitten with Lord Voldemort.
After escaping Azkaban, she likely lived with her sister until her death.
As mentioned in the question, Bellatrix’s love life is not detailed in the seven books, so it would not be possible to address where she lived “after she left her husband” since it is not known if she ever did to begin with. However, the question of where she lived after escaping Azkaban until her death in battle is more answerable.
It seems likely that she went to live with her sister Narcissa soon after her escape. At the time Narcissa seeks help from Snape, she seems to have been around her sister quite a lot leading up to that decision, and spent much time convincing her to not go through with it. Presuming this indeed means she was living at Malfoy Manor very shortly after her escape, it would solely be because of her sister living there - Voldemort most likely did not begin living there until the seventh book, so her choice would not have been on account of his presence.
“Cissy – Narcissa – listen to me –’
The second woman caught the first and seized her arm, but the other wrenched it away.
‘Go back, Bella!’
‘You must listen to me!’
‘I’ve listened already. I’ve made my decision. Leave me alone!”
- Harry Potter and the Half-Blood Prince, Chapter 2 (Spinner’s End)
While it is not confirmed who taught Draco how to do Occlumency, Snape suspects that it is Bellatrix, which means Snape had reason to believe that she was around Draco enough during this time for this to be possible.
“There was a pause and then Snape said quietly, ‘Ah … Aunt Bellatrix has been teaching you Occlumency, I see. What thoughts are you trying to conceal from your master, Draco?”
- Harry Potter and the Half-Blood Prince, Chapter 15 (The Unbreakable Vow)
During the Death Eaters’ meeting once Voldemort had taken up residence at Malfoy Manor, Bellatrix refers to it as her family’s home, showing she feels at least some right to be there and connection to it. She still lived there then, and though Voldemort being there does not bother her, it is not the reason she lived there as she had been there before him.
“My Lord,’ said a dark woman halfway down the table, her voice constricted with emotion, ‘it is an honour to have you here, in our family’s house. There can be no higher pleasure.”
- Harry Potter and Deathly Hallows, Chapter 1 (The Dark Lord Ascending)
When Voldemort went to Nurmengard in search of Grindelwald, Bellatrix still lived at Malfoy Manor.
“The drawing-room door opened behind Harry. A woman spoke, and the sound of the voice wound Harry’s fear to an even higher pitch.
‘What is this? What’s happened, Cissy?’
Bellatrix Lestrange walked slowly around the prisoners, and stopped on Harry’s right, staring at Hermione through her heavily lidded eyes.”
- Harry Potter and the Deathly Hallows, Chapter 23 (Malfoy Manor)
After Harry, Ron, and Hermione escape it, it is rumored that Bellatrix and the Malfoys are confined to Malfoy Manor by Voldemort.
“Well, I confess I am surprised to see you out and about, Bellatrix.’
‘Really? Why?’ asked Hermione.
‘Well,’ Travers coughed, ‘I heard that the inhabitants of Malfoy Manor were confined to the house, after the … ah … escape.”
* - Harry Potter and the Deathly Hallows, Chapter 27 (Gringotts)*
It is very soon after this that Bellatrix fought and died in the battle at Hogwarts, so she would not have had time to change residence after this exchange.