Theresa May reaches out to her rival in a bid to break Brexit deadlock

U.K. Prime Minister Theresa May and the leader of the main opposition party, Jeremy Corbyn, meet on Friday for the third consecutive day of talks in a bid to break the current Brexit deadlock.

The two leaders, who have fought bitterly for months over Britain's future, are now trying to agree a deal that can then be negotiated with the rest of the EU.

Earlier this week, May outlined to Parliament the terms for the talks, promising to support the final outcome and take it to Brussels. "If we cannot agree on a single unified approach, then we would instead agree a number of options for the Future Relationship that we could put to the House (of Commons) in a series of votes to determine which course to pursue," she said.

"Crucially, the government stands ready to abide by the decision of the House. But to make this process work, the opposition would need to agree to this too."

Unlike in the U.S., the party that controls the legislature (Parliament) in the United Kingdom also controls the executive (government).

Therefore, for Corbyn to be engaged in days of talks with May to resolve an issue is yet another example of how Brexit has shaken British politics.

The problem for May, and the reason why she has been forced into talks, is that she cannot rely on the unanimous support of members of her own Conservative Party who are split between different versions of Brexit, and even no Brexit at all.

Corbyn and Labour have the same issue of division. Many of Labour's lawmakers in the north of England secured their seats in Parliament in areas which voted to leave the European Union. Many southern seats won by Labour, especially those in London, voted to remain.

One big area of contention between the two leaders is whether the U.K. should remain a member of the customs union. This arrangement allows for members of the European Union to set common tariffs on goods moving within the trading bloc.

It's considered a barrier for setting up new trade deals with countries outside the EU and May has previously deemed it a "red line" issue for her, stating the U.K. must leave the arrangement. The Labour leadership, however, wishes to remain in the customs union, arguing it would protect U.K. businesses and jobs.

For any deal to emerge from the talks, May would likely have to break this previous promise and add the customs union to her withdrawal deal.

If agreed, such a deal would put pressure on Corbyn too. It would end freedom of movement into the U.K. from continental Europe and of course from the U.K to the EU member states. Many within the Labour party argue that stopping migration would simply be helping Downing Street to deliver a "Tory Brexit."

More pressure is also being placed on Corbyn to attach a second referendum to any cross-party arrangement. His own deputy, Tom Watson, is spearheading a public drive to put any agreed deal up against the choice of not leaving the EU at all.

Corbyn has publicly pressured May to state that she would support the will of Parliament on Brexit, suggesting that he himself would also support what lawmakers can find a majority on.

But while a deal cooked up between Britain's two biggest parties may sound like sensible politics, many consider the chances of May and Corbyn emerging hand-in-hand as slim to none.

May, while often inscrutable, has the voting record of a center-right politician that supports private enterprise and seeks toughness on immigration and crime. At the other end of the political spectrum, Corbyn is viewed as a hard-left unionist who would take key businesses and sectors into state control and raise taxes on corporations and high earners.

Additionally, many Labour supporters also suspect May has no real interest in compromise and has reached out to Corbyn in a bid to "dip his hands in the blood" of her party's Brexit mess.

If agreeing a deal looks tricky enough, then passing it through their own lawmakers is also far from guaranteed.

Although there are 650 seats in the House of Commons, seven lawmakers from the Northern Irish nationalist party Sinn Fein do not attend, and by convention the house speaker and his three deputies do not vote. This means that the government needs 320 votes for a simple majority.

The Financial Times reported Thursday that only around 185 Conservatives and 145 Labour lawmakers are thought likely to vote with their leaders. Based on those numbers, a majority of 10 ten votes is achieved.

But if such a slim mandate can be achieved, May would then head to Brussels hoping to convince European leaders that Britain's terms of leaving Europe were now in place.

Now watch: Enrico Letta: More countries will want to leave European Union soon

Read More