Maggie Gyllenhaal Michael Friberg/Contour by Getty Images

Maggie Gyllenhaal is one of the great cryers on screen. In her Oscar-nominated role in "Crazy Heart," or as a mother just out of prison in "Sherrybaby," it seems her eyes fill effortlessly and soon she's sniffling and wiping away tears. She has especially harrowing reasons to weep as Nessa Stein, the British heroine of the eight-part series "The Honorable Woman," a family drama wrapped in a political thriller about the Middle East.

As children, Nessa and her brother were witnesses when their father, a tycoon who sold arms to Israel, was murdered by a Palestinian. In the present day, Nessa has been made a Baroness and runs the family company, now laying communications cables between Israel and Gaza, her attempt at encouraging peace. Hugo Blick, the series' writer and director (known in the U.K. for his crime-themed miniseries "The Shadow Line" and in the U.S. not at all), creates an international landscape of secrets and betrayals, back-channel deals and kidnappings, with tentacles reaching MI6 and the CIA. But the story never strays far from Ms. Gyllenhaal as Nessa, whose personal life mirrors the volatile politics of the region.

(Read Dorothy Rabinowitz's review of "The Honorable Woman.')

In person, Ms. Gyllenhaal has the same elegantly casual style as Nessa, but she laughs a lot more. Her career has been a deliberate mix of arty little indies (she is a sullen theremin player in the coming "Frank"), midsize movies and the occasional blockbuster, like "The Dark Knight." She has appeared off-Broadway doing Chekhov with her husband, Peter Sarsgaard, and this fall will appear on Broadway, starring in a revival of "The Real Thing" by Tom Stoppard. "The Honorable Woman," complete with Ms. Gyllenhaal doing a precise British accent, begins on SundanceTV on July 31. Edited from an interview:

In one episode, there's a montage of news footage from the Middle East over the decades; it looks exactly like the news from Israel and Gaza today. Does that surprise you?

From left, Eve Best, Tobias Menzies, Maggie Gyllenhaal and Genevieve O'Reilly in a scene from 'The Honorable Woman.' SundanceTV

When it was written and when it was shot [last year] that area of the world was in a relatively calm moment. But you would have to have your head in the sand not to know that underneath that calm was an imminent possibility of violence again. I hope that the show will provide at least an hour once a week for eight weeks to think about what's happening and to feel about what's happening in that part of the world. It doesn't take a stand, but it lays out compassionately, empathetically, intelligently at least some aspects of what's happening over there. And it asks the audience to think and feel for themselves about it, which is exactly what's needed just now.

The Israelis, the Palestinians, the British, the Americans—no one comes out looking very good. There are deceptions, double-crosses, triple-crosses.

But why? Why all this deception and lack of trust and confusion? It's an inheritance of pain from all sides.

Nessa gives a public speech at the start and says that Palestinian poverty breeds terrorism, and economic opportunity for all makes Israel more secure. Do you agree with that?

I want to keep what I feel out of it because the piece articulates, so much better than I can, something that's hard to articulate in a couple of sentences.

Then do you think Nessa's statement is naive or idealistic?

I think most people who are after fundamentally shifting something, really changing something, have an aspect of them that is naive. I certainly do; I relate to that. And I think Nessa does too. She's also obviously extremely savvy. I think she understands, in a way that I am working to understand right this minute, that standing up and saying 'This is what I think and you can take it or leave it!' is not ever going to be effective in terms of opening someone's mind, and I'm trying to learn from her. She's the opposite of naive in that way.

Do you think you've made mistakes like that in the past? When you spoke against going into Iraq over a decade ago, were there any repercussions?

Ms. Gyllenhaal in 'The Honorable Woman' SundanceTV

I think there was a kind of energy at that time to scare people out of saying much about anything, so I thought there were going to be repercussions and it scared me. I was around 25. But there weren't. It was really interesting to note: Nothing happened to me. But if my intention at the time was to make someone who thought differently than I did understand what I wanted to say, I absolutely failed. What's the point of yelling out your point of view about something if it's just to have other people who agree with you pat you on the back?

How does that knowledge inform "The Honorable Woman"?

Everybody knows that there are people on either side of this conflict with a vise grip on their point of view, and it's so easy for a conversation to shut down when someone has that kind of position.

I've been trying at this moment to read everything I can on both sides: pro-Palestinian pieces that are written by intellectuals, and pro-Israeli pieces written by intellectuals. Now those are polarizing but what I'm trying to do—and honestly it's the same challenge I'm asking the audience to do—is read what's on either side and come up with my own point of view, then keep it to myself and let people watch the show. Behind my intention in making this is compassion and, maybe it's naive, a belief in the possibility of reconciliation, which our show never takes off the table.

About the title: Is she honorable? What does honor even mean in this world where everyone is so duplicitous?

In the U.K. the title also refers to her being made a baroness; it doesn't play quite the same way here. I certainly always took it as an American mind, thinking: what does it mean to be honorable? I think there's almost a question mark at the end of the title. And do you have to come down hard-core on one side or the other, she isn't or she is? Are you an honorable woman? Am I an honorable woman? I'm giving it my best shot, I certainly am trying. And I think that's the same with Nessa. For her, what's honorable shifts massively.

At the start Nessa has an incredibly calm demeanor, almost regal, very controlled. But when she falls apart she really falls apart. How did you decide to play her so controlled?

I didn't decide anything really. The experience that Nessa has often mirrored the experience I was having as an actress. For instance, that first speech—this incredibly eloquent, graceful, funny speech that Hugo wrote. I had the challenge of giving that speech however many times, in another accent, standing at this podium in front of all these people, trying to make the argument for what she believes. There was some part of me that wanted to do it remarkably, that was pushing aside the part of me who's fallible, who makes mistakes, who's scared. Which is exactly what Nessa's doing: Nessa wants to be remarkable, she wants to be extraordinary. Basically you walk into the story at the moment where she can't keep it up anymore. So she gets through that speech and later falls apart in her friend's arms. She can't speak a word, she starts weeping. I didn't know I was going to start weeping.

She's also self-destructive at times, as if she feels she needs to be punished. Why?

I think it has a lot to do with who her father is. It's very, very complicated but I think she inherited something from her father that she has had to contain, and she feels very guilty about it. So many things in her life are a response to him having been an arms dealer and an incredibly productive one, who probably helped to kill thousands and thousands of people.

Isn't your husband writing a film for you?

Yes, and we're hoping he's going to direct it. He's about two-thirds through the script. We want to make it super super low-budget. It takes place in 1904. We want to shoot maybe 10 days in upstate NY and then three weeks in Northern Canada. And then I'm in the process of buying [the rights to] a book I can't tell you about yet. So much of the time I'm just raising my hand and saying 'I'd like to do that,' or waiting for someone else to write something. It's nice to be able to try to put a budget together.