I know that you are supposed to separate dependent clauses and introductory phrases with a comma if they start the sentence. Examples:
If you are ill, you ought to see a doctor. (dependent clause)
After the adjustment for inflation, real wages have decreased while corporate profits have grown. (introductory phrase)
Reference: https://owl.english.purdue.edu/owl/resource/607/02/
However, the above reference does not give examples of punctuating compound and complex-compound sentences that include dependent clauses and introductory phrases in the second part of the sentence.
Example of such a sentence with original punctuation kept (I took it from Wiktionary — an example of usage for word particular in sixth definition):
He is very particular about his food and if it isn't cooked to perfection he will send it back.
I think that there should be a comma after the word food, but that is not my question.
My question is if there should be a comma after the word perfection.
Similarly, sould there be a comma after an introductory phrase which comes after the first indepedent clause but before the second indepedent clause? (Although it is probably considered to be a part of that second independent clause.)
[independent clause], + coordinating conjunction + [introductiory phrase] + [independent clause].
Comma or no comma after an introductory phrase in the above structure?