Meanwhile, a statement like My weekly supermarket bill has been going up since the Brexit vote is completely meaningless.
By definition, it is anecdotal. A single piece of price data observed only by you with no evidence is not proof that EU grown vegetables are more expensive than those from Mexico.
To say that someones supermarket bill going up is somehow more anecdotal than "I found asparagus from Spain that cost more than asparagus from Mexico" is absurd.
Empirical evidence in this case would be a large scale survey of the prices of comparable vegetables across multiple supermarkets, over a long period (several years to cover seasonal variance in prices).
You have pointed out one good amongst thousands that was cheaper when it came from outside the EU. I'm sure there are many others but that's irrelevant. Those specific goods does not mean that this is the result of the EU. What everyone is afraid of is what will happen if tariffs are applied to EU goods when they come into the UK.
Time for anecdotal evidence. My weekly supermarket bill has been going up since the Brexit vote.