Add carriage return to output of `tail` while using `grep`

I'm not sure what you want, but awk can probably do it easily. To add an extra newline after each matching line:

tail -f production.log |
awk '/Processing|compatible;|Completed in / {print; print ""}'

To add a newline between all blocks of non-consecutive matching lines:

tail -f production.log |
awk '
/Processing|compatible;|Completed in / {
    if (NR == n+1) print "";
    n = NR;
    print;
}'

grep will filter out empty lines since they don't match any pattern. In the case empty line separators appear in the original file you can let them pass through, adding -e "^$".

If those empty lines are not present in the original file, you'll have to add them. This is an example of how to do it:

 tail -f production.log | grep -e … | sed 's/\(^Processing \)/\n\1/'

Edit: if you want an empty line after every other line use sed 's/$/\n/' instead. And of course you can add as many \n as you like.