Not a clinical study, but there is data coming out of Israel that isn’t being discussed here at all, possibly because it’s released in Hebrew. Here’s a good breakdown of delta in Israel.
And the relevant portion on natural Immunity.
Who isn’t getting Delta?
Well, it’s hard to say for sure because of confounders in the data, but it looks like recovered people — people who had tested positive for the coronavirus in a past PCR test — are massively under-represented. Recovered people are around 9% of Israel’s population, but they’re less than 1% of current cases.
That has led to speculation that recovery from past Covid is more protective against infection from the Delta Variant than vaccination. Known recovered people in Israel are mostly not vaccinated, though some chose to get a single booster dose
There could be other explanations for this result. The Delta wave is, for now at least, not affecting the communities hardest hit by Israel’s previous Covid waves: Ultra Orthodox Jews and Israeli Arabs. So perhaps recovered people aren’t being exposed to the virus yet.
But I’m not sure that’s enough to explain the sharp difference. From data I was passed last week, recovered people were testing at only 0.1% positivity, compared to more than 1% for vaccinated people.