In the lobby of a Dublin hotel, Charles Thompson IV, better known as Pixies frontman Black Francis, is literally acting out bassist Kim Deal's recent departure from the band for my benefit. "So," he says, walking away from the table at which guitarist Joey Santiago and drummer David Lovering are also seated, "you gotta imagine we're in this coffee shop in Monmouth. Not Costa, the other one, with the Italian name."

Caffè Nero? You're saying that the original line up of Pixies – arguably the most important and influential American guitar band of their era; purveyors of howled songs about incest, lust, Biblical violence, death and UFOs; a band who, on arrival, appeared so strange and alien to British audiences that, as one critic put it, they "seemed to have crawled out of the desert, babbling in tongues" – lost their bassist in a Caffè Nero in a market town in south Wales?

Not for the last time during our meeting, Black Francis frowns and nods briskly, in a way which suggests that something I find a bit peculiar doesn't seem particularly peculiar to him. "Right. Caffè Nero. So we're sitting there in chairs chatting, and Kim walks in, like this" – he walks back to the table, wreathed in smiles – "'Hey guys, how's it goin'? Good to see you guys.' We're talking about a football game or something and she's joining in – 'Oh, did you see that?' – and then she just said, 'So, I'm flying home tomorrow.' That's my memory of it anyway. And I think that Joey and I said: 'OK,' and we just kind of walked out."

Hang on, the woman you just said was "practically the mascot of the band" quits halfway through the recording sessions for your first album in 22 years (the songs are being released online, without warning, in batches of four – the first, EP1, came out in September – but it's still an album in all but name) and you didn't ask her why? "Well, I think I probably sensed that it was going to get tense," shrugs Francis, "so I left."

"And I had an errand to go on, actually, I had to get a slide for my guitar," says Santiago, "so I followed him. The conversation was over."

"It was classic Pixies," nods Francis.

"I talked to my better half," says Santiago, "and she went: 'Why didn't you just talk about it?' I go: 'What? What the fuck are you talking about?' 'Well, why didn't you just sit down and talk about it?'" He laughs. "Why would we do that? It makes too much sense. We're shitty communicators."

It does all seem a little odd, which, you could argue, makes it entirely of a piece with pretty much everything else about Pixies. Their initial rise to fame was, they concede, strange. Within 18 months of their first rehearsal, they had developed a huge, obsessive European following while back in the US they had barely played outside their hometown, Boston. After a German gig on their first European tour, Santiago remembers a mob of fans attempting to pull him out of the door of the band's van by his feet.

They appeared at the height of an extraordinary era for guitar music – their breakthrough album Surfer Rosa came out in the same year as Sonic Youth's Daydream Nation, Dinosaur Jr's Bug and My Bloody Valentine's Isn't Anything. But Pixies seemed somehow removed from everything else that was happening: they were never part of a scene, they appeared to exist entirely in their own world, and it was hard to identify their influences. They looked nothing like anyone's idea of a cool indie band, something that holds true today: Black Francis arrives at the interview wearing what appears to be an embroidered poncho.

Even the circumstances of their reformation in 2003 were unlikely. Attempting to parry yet another question about whether Pixies would get back together during a radio interview, Francis jokingly quoted George Harrison's mischievous line about how the Beatles might secretly have reformed and not told anyone: "There's stuff that goes on that you guys don't know about." But nobody got the reference and "the next day it was literally on CNN". Francis found himself fielding calls from the other former Pixies asking him what the hell was going on.

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All three remaining members of the band appear genuinely surprised that Deal left, even if the news wasn't exactly greeted with shock by the outside world, the tension between Black Francis and Kim Deal being one of the facts about Pixies that anyone with even a passing interest in the band knows. It's up there with the vast, unimpeachable reputation of the five albums they made between 1986 and 1991, Kurt Cobain claiming Smells Like Teen Spirit was his attempt at sounding like them, Radiohead refusing to go onstage after them at a 2004 festival because "it would be like the Beatles supporting us", and their 1988 track Where Is My Mind? soundtracking the climactic scene of Fight Club.

The rivalry first erupted during the making of their third album, 1989's Doolittle, and ultimately led to their initial break-up in 1993. Francis wrote the songs; the band declined to play Deal's, deeming them "way different than our other stuff". He was the frontman, but she swiftly became the band's most celebrated member, exuding a nonchalant, cigarette-smoking cool onstage, her calm, measured voice the perfect foil to Francis's yowling and screaming: in the 2006 Pixies documentary, loudquietloud, Deal is pursued by her own coterie of fans, who hyperventilate when they meet her and hold up signs during gigs proclaiming her to be God.

As such, her disappearance casts something of a shadow over the band. There are Pixies fans who don't think it's really Pixies unless Deal is stage right, fag smouldering in gob: it's hard not to think that EP1 – which, as Lowering notes, got "bad reviews, which we've never really had" – might have been more favourably looked upon if Deal was still in the band.

For that reason, I had expected the band to be wary of discussing her departure, but it's quite the opposite. They are not, it has to be said, the most relaxed interviewees in the world: at one point, I notice Santiago snapping a rubber band against his wrist, in the manner of someone who suffers from anxiety. Inquiries about Francis's songwriting processes or the psychological effect of playing the same album, in order, at every gig for two years – as Pixies did with Doolittle between 2009 and 2011 – don't yield a great deal, beyond Francis wryly comparing the latter to working in a restaurant: "I want that macaroni cheese and I want it exactly the way you made it before." Mention Deal's name, however, and Francis is out of his seat and doing impersonations of the moment she left. I get the impression they actually rather like talking about her departure, because they are still trying to puzzle it out themselves (Deal has remained silent on the subject).

They seem pretty certain that, at its root, was the question of whether or not a reformed classic band should record new material, an idea to which Deal remained vocally resistant. "Some of her reasons were perfectly valid," Francis says. "Just sort of like, if it's not broke … We're getting requested to do reunion tours, we've got the reputation, we've got these five albums, that's what we're known for. It's a risky move when bands get back together and make new songs, it's so hard. You know, I validate her for her reasons. But ultimately, you know, come on. I'm not saying she intended it in this way, but if you're sensitive, and I like to think maybe I am a little sensitive anyway, you're going to take that as a sign of non-validation from somebody else. They're saying: 'Well, look, you did some good work when you were young, but I don't trust you now to come up with the kind of fodder this band needs. You don't have it any more.'"

Besides, Francis says, once their reformation had gone on longer than their initial career, the rest of the band were starting to feel wary about just playing the old material, particularly when they found themselves booked to play a Canadian casino, the kind of venue that is traditionally the preserve of oldies acts: "It was just sort of symbolic, like ha-ha, here we are, at the casino. Uh-oh, we're in Palm Springs." They tried everything to convince her, he says. "We got together in LA without her, just to see what we got, like we could seduce her in the process, come up with something that would tickle her ears and she'd go: 'Oh wow, you guys are really up to something good here'. Then I had the bright idea of, like: 'Let's get together just like in the old days, just the four of us in a rehearsal room, jamming and rocking out.' We got a rehearsal space in the same damn area of Boston where we started. It was disastrous. From the very first day, every day."

"Passive aggressiveness," nods Santiago. "People think we shout at each other, but no. It was just the worst three fucking weeks of my life. Really it was, it couldn't get any worse than that."

"Three weeks?" frowns Francis. "We were only there a week."

Nevertheless, by the time they arrived at Monmouth's famous Rockfield Studios, with Doolittle producer Gil Norton in tow, Deal appeared to have come around to the idea. The sessions went well. The night before Deal announced her departure, the band went out to dinner together. "We had a nice meal, a nice chat," says Francis. "I can't remember the last time we had a meal together, just the four of us, without our manager or our crew. Afterwards, I realised, oh, that was like the Last Supper, that was the goodbye dinner."

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They briefly considered breaking up, he says. "We didn't know what to do for maybe a day, but then we were like: "Well, this studio's booked for another six weeks. Jesus Christ, we're going to fucking record these damn songs. Fuck it, we're going to do it. So we followed through. Even with her leaving, which was sort of like the big no-no, you know, no one wants her to leave – 'Oh God, not Kim Deal, anybody, but not Kim Deal' – we still went: 'No, we're going to finish the job.'"

So they carried on, with friends filling in: Kim Shattuck of LA pop-punk band the Muffs is currently playing bass. Their European tour, which finishes in London on Monday, is sold out and there seems to be a general consensus that Pixies, who suddenly find themselves with everything to prove, are playing ferociously. "And from my perch on the drums," smiles Lowering, "I've noticed that not too many people have been leaving to use the restroom during the new songs."

They are pleased with the new material, the novelty of bad reviews notwithstanding. In any case, it's not like Francis didn't anticipate the reception: "Well, looksee what the wind blew back," he sings on EP1's Indie Cindy, "as we follow the bouncing ball, they call this dance the washed-up crawl". In fact, he says, he struggled for a while to write songs for Pixies again, bothered by what Indie Cindy refers to as the band's "more or less unchequered record", until Gil Norton came up with the unlikely suggestion he simply imagined the band had never broken up, but instead spent the intervening years touring in outer space.

"We were elsewhere, off the planet, sequestered in space for 20 years. Still making music, still being who we are and now we're back: what does that sound like?"

Santiago and Lovering protest they had no idea what Norton was on about, but Francis really ran with the idea: painting gig posters for intergalactic Pixies shows, inventing a planet called Pan, on which the band had apparently been particularly big, famed for their appearing at a venue called the Panoprodomo. "It's had an impact on some of the songs, you know: 'Oh, I've got a new one here, what do I write about? OK, space opera! Here we go!'"

Panoprodomo and passive aggression, sudden unexplained departures and space operas. Your band is genuinely weird, I say. Black Francis frowns. "How so?" he asks, apparently surprised.

Pixies play Hammersmith Apollo, London W6, on 24 and 25 November.