Henrik Ibsen's A Doll's House is an example of realism, although the terms "realism" and "naturalism" are often used interchangeably.
A distinction between realism and naturalism is that naturalism is hyperrealism, a heightened form of realism. Naturalism is realism taken to the extreme.
Realism has one foot in theatricality and reverts to theatricality when it serves the play (and the playwright), whereas naturalism has walked away completely from theatricality.
A realistic play like A Doll's House retains, and is dependent on, conventional theatrical form and structure. A Doll's House has exposition, rising action, complication(s), climax, and resolution, as well as plot, characters, and a theme.
A naturalistic play doesn't rely on theatrical conventions in the same was as a realistic drama does. A naturalistic play doesn't rely on conventional play structure and might not have a plot, characters, or a theme.
A naturalistic play presents what's called "a slice of life," a term that originated around 1890 as "tranche de vie" by French playwright Jean Jullien (1854–1919). It's as if two hours of a certain number of people's lives were lifted directly from their life and put down on a stage wholly intact in every way.
An extreme example of naturalism would be a play in which characters walk onto the stage as if they just walked off the street or came in from another room, and they walk into a set that is essentially perfect in every way, down the smallest detail. Everything on the set is absolutely real.
The characters engage in informal, colloquial conversation, with all the pauses, repetition, "uhm"s, "ah"s, and "you know"s, irrelevant stories, and meandering flow of words found in normal conversations. The conversation touches on subjects which might or might not be related to the central issue of the play (assuming the central issue can be discerned from the conversation).
The action of the play, if any, or, more likely, just the conversation, occurs in real time. Stage time equals real time. Time is not expanded or contracted as it is in A Doll's House.
Often there is no discernible cause-and-effect relationship between events in the play.
There might or might not be character development. Characters don't necessarily change or "go on a journey" from the beginning of the play to the end, discovering things about themselves or each other that are meaningful to themselves, each other, or the play as a whole.
Nora goes on a journey. Her character changes from the beginning of the play to the end of the play. Interestingly, her husband Torvald is almost naturalistic in that he doesn't go on a journey. He doesn't change.
Two or three hours later, the characters leave the stage, often without any conflict reaching a noticeable climax or any resolution of the central issue (if any) being achieved.
A realistic play, like A Doll's House, looks real, or it looks real enough for the purposes of the play.
A naturalistic play is real, or as close to real as a theatrical presentation can get.