The Manchester United manager came onto the Amex pitch with a bottle of red after Ince said it was a 'lack of respect' not being invited for a drink while managing Reading.
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Ince left the Royals on April 11 after a poor run of form, but during his stint he faced United in the FA Cup earlier this season.
Reading lost 3-1 on the day, and Ince criticised Ten Hag afterwards for not inviting him to 'pop in for a drink'.
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Now working as a TV pundit, Ince was approached by the United boss with a bottle in hand, to much laughter.
"I owe him one!" Ten Hag said. "There's not always time for a drink after the game but you're always welcome."
Ince replied: "That's so kind of you, thank you very much. We'll share it after you win the cup final."
Ince played for United between 1989 and 1995, and expected more of a reception when he faced his old side with Reading.
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Speaking after the February FA Cup fixture, he said: "I went straight up to see Sir Alex [Ferguson] because I got a text from him afterwards to say 'come up and see me'.
"So myself, my wife and Thomas [Ince] went up there and spent an hour with him. It disappointed me because none of the United staff asked me in for a glass of wine. I didn't like that, it's a lack of respect.
"Whoever it is, win, lose or draw, you say pop in for a drink. Whether it's wine or tea it doesn't really matter.
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"None of their staff did that so that was disappointing. I couldn't give a ms but I just pride it on myself, as a manager, whether or not you get beaten, you say come in for a glass of wine."