What is realistic dialogue? That's hard to say as to be brutally honest it's a very subjective notion. What I think is realistic dialogue may not be what the next person thinks it is.
The most obvious starting points are accents and slang. Should the writer attempt to introduce these two verbal notions into dialogue? Most of the advice given to me is 'no'. Accents and slang put the reader off because they require deciphering, can date the story, and can be done extremely badly. Using accents and slang can add realism but can also marginalise your audience. Writers tend to want to reach the widest audience they possibly can and to do that communication needs to be clear.
The other argument is that dialogue has to be true to the character. You can't have an eleven year old boy talking like an elderly grandfather about philosophy or a Queen gossiping like a public house landlady. The character dictates the dialogue in the same way character dictates behaviour. I think as long as the writer follows that rule then there shouldn't be an issue of the dialogue being unrealistic.
I've also been given the advice that the more you listen to people the more you can pick out exactly how they speak. A man will speak one way, a woman another, a child another, a drug addict another, etc etc. This is assuming that this dialogue is unique to 'types' of people and can be recognisably reproduced ie stereotypical utterances by stereotypical people. This theory seems suspect to me. Stereotypes are not real people in all their nuanced glory so how can stereotypical dialogue be realistic dialogue?
To a certain extent listening to people allows a taste, a flavour of how dialogue works and adding that flavour makes dialogue better. But dialogue is constructed around a story, it is essentially artificial, an imagined reaction to an imagined situation. You can't sample that off the street, you have to make it up. A writer has to measure their own reaction and subvert it into their character's reaction and translate that into dialogue.