My theory is that when we have an internal imbalance, the body is trying its best to get rid of all of the toxins. When these toxins come out of our pores I think they can begin to smell really bad. I also believe this is a way our body uses to signal that something needs to be taken care off. After all, if it didn't smell bad, you wouldn't be here posting about this warning.
After many years of a highly processed diet, too many sweets and not enough fruit and veg, my scalp began to smell really bad regardless of how many times I washed my hair or what I used (Smelly Hair Syndrome). After a while I began looking 'on the inside' and this is why I have that theory.
I no longer have scalp odour and I got rid of it by improving my diet (more fruit and veg), taking antifungal supplements for a few months with Probiotics. Then I stopped all antifungals and continued with Probiotics along with eating better in general.
In your case It could be your diet is not alkaline enough but I would need to know more about what you eat to be sure. All of the above is simply my unprofessional theories and I have no health related qualifications.
However, if the odour is due to your diet then should be able to be bought back to balance.
I wouldn't say that 'leaky gut' causes bad body odour directly, rather that having it means your internally out of balance and unable to rid yourself of internal toxins and pathogens efficiently which could potentially lead to bad odour as your body struggles to get rid of the toxins.