A past member of the UK’s monetary policy committee once told me that they got much more intelligent questions from committees of the House of Lords compared to committees of the House of Commons. This should not be too surprising, as there are some people with considerable knowledge and experience in the Lords.
“The notion that a country can have complete regulatory sovereignty while engaging in comprehensive free trade with partners is based on a misunderstanding of the nature of free trade. Modern FTAs involve extensive regulatory harmonisation in order to eliminate non-tariff barriers, and surveillance and dispute resolution arrangements to monitor and enforce implementation. The liberalisation of trade thus requires states to agree to limit the exercise of their sovereignty. The four frameworks considered in this report all require different trade-offs between market access and the exercise of sovereignty. As a general rule, the deeper the trade relationship, the greater the loss of sovereignty.”
There you have, in one calm and measured paragraph, the contradiction at the heart of the argument put forward by Liam Fox and others that leaving the EU will allow the UK to become a ‘champion of free trade’. You cannot be a champion of free trade, and have sovereignty in the form of taking back control.
It is not a contradiction, of course, if you are happy to accept the regulatory standards of the US, China or India. That appears to be the position of Leave leaders like MP Jacob Rees Mogg. Ellie Mae O’Hagan spells out what this may mean in practice. Lead in toys - bring them in so we can sign a trade agreement with China. And you can be sure that this will be the nature of the discussion every time a trade deal is signed. In each case we will be told that we have to accept this drop in regulatory standards, because British export jobs are on the line.
This is the point of Dani Rodrik’s famous impossible trilemma: you cannot have all three of the nation state, democratic politics and deep economic integration (aka free trade). His trilemma replaces sovereignty, by which in meant in this context the nation state being able to do what it likes, by democracy. In the past I have always found this problematic. Surely a democracy can decide to give away a bit of its sovereignty in return for the benefits of international cooperation (in the form of trade deals, or indeed any other kind of international cooperation). After all, every adult in a relationship knows that this relationship means certain restrictions on doing just what they would like.
At first sight, it would seem as if the Brexit vote shows Rodrik is right. Democracy voted to take back control, which means reducing trade integration. But I think it is becoming increasingly clear that this is the wrong interpretation. Voters were told they could take back control and be no worse off, and polls make it clear that message was believed by many Leave voters. As it becomes clear that people will be worse off, as depreciation induced inflation cuts real wages, opinion is changing. Polls already suggest that if the vote was held again, we would get a different result. Polls also suggest more voters want to prioritise favourable trade deals in negotiations, not curb immigration. Over the next year or two this will only intensify, as prices rise, as companies make plans to leave the UK, as the problems caused by declining immigration emerge, and as the UK’s weak negotiating position becomes clearer.
Leavers know this, hence the attempts to remove any kind of democratic oversight from the Brexit process. At present MPs appear transfixed by the light of the Brexit vote, even though most know Brexit is an act of self harm. But it was utterly predictable from the day that Corbyn was re-elected that we would see a revival of the Liberal Democrats. As they chalk up election victories, it might just be possible that we could yet see some democratic oversight of the Brexit process. 
To see a model of what could and should happen, look to Switzerland, where referendums are part of political life. In February 2014 Switzerland voted to restrict immigration from the EU, even though this jeopardised their trade relations with the EU. Since then the EU has insisted that its bilateral trade deals with Switzerland depend on free movement. As a result the Swiss parliament has backed down, and just passed measures which greatly diverge from the referendum proposal, even though in Switzerland referendums are (unlike the UK) meant to be constitutionally binding.
So we see in Switzerland, and perhaps we will see in the UK, that parliamentary democracy can be compatible with trade integration. Rodrik is right that deep trade integration (the pressures from which will continue, as Richard Baldwin outlines) puts pressure on the ability of nation states to decide on their own laws, but a democratically negotiated compromise is possible. (After all, national languages are a barrier to trade.) Perhaps the examples of the UK and Switzerland suggest two things: first that these negotiated compromises should be out in the open rather than done behind closed doors, and secondly that what is very difficult to mix are trade integration with national referendums.
 The conventional logic is that MPs would go along with Brexit because the Remain vote is concentrated in too few constituencies. But outside Scotland the Leave vote is now split three ways (Conservative, Labour and UKIP). Labour may mock the LibDems as Brexit deniers, but as the LibDems get the votes not only of people who deeply care about being part of Europe (see the surge in Remain identity noted here) but also of those that voted Leave but are now getting concerned, they will be the ones with the last laugh.