"The one from Kcott "almost" worked. The double-quote fields are correct, the ones before the quotes are missing :)."

Yes, I forgot to capture the first part of the strings. Fixing that, and then making changes for your altered input and updated spec:

$ perl -pE 's/^([^"]+")([^"]+)/$1 . $2 =~ y{\t}{,}r/e'
a b "x1 x2"
a b "x1,x2"
c d "x2"
c d "x2"
e f "x3 x4      x5"
e f "x3,x4,x5"
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"I was not able to add "tabs" in the example ..."

Surely you must mean something else. I pressed the key labelled "TAB" on my keyboard to add them to my input. Admittedly, it may not be easy to see the difference between a tab and a space, but the output gives it away:

$ perl -pE 's/^([^"]+")([^"]+)/$1 . $2 =~ y{\t}{,}r/e'
a b "x1 x2"
a b "x1 x2"
[download]