The argument that leaving the Customs Union will lead to a reduction in the UK’s food prices seems to originate in a Policy Exchange think tank paper. But the “evidence” for this is a list of EU WTO tariffs on agricultural products. policyexchange.org.uk/wp-content/upl…
Here’s where the UK imports fruit from, the EU (and that includes us) has tariff-negating trade deals with pretty much all these countries (except Brazil). 
And are our food prices more expensive than comparable countries round the world? Not if you look at meat. UK ranks well both on price and affordability, despite not being a major producer. This is ranked on price of chicken breast.
UK comes third in global food security ranking, and on affordability we again rank highly, despite not being a major food producer and relying on imports.
So as we don’t apply WTO tariffs on all the food we import, and our food prices are already very competitive, where is the evidence that leaving the Customs Union will reduce them? Even if we import chlorinated chicken from the US, it’s not going to make our meat cheaper.
