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Cracks within the "field" of a tiled area often point to a poorly set individual tile (lack of adhesion and tile flex) or to more wholesale type of issues such as substrate flex, lack of or insufficient or poorly installed backer. If it is just a bad tile or two this can be repaired fairly
Simply using a grout scraper remove the stuff on the corners and use a high end matching color grout caulk that can be purchased any any good quality tile store. This material will flex and bend resisting the cracking that your now seeing.
For the corners get a grout "saw" and clean out the trouble areas..then apply some matching caulk.
Pat - rope putty? do you mean the foam backer rod or something else? Never heard of rope putty. I'm assuming something else as you wouldn't paint backer rod. What is it and where'd you get it?
If the shower base is concrete or tile then there could be a fitting leak or a membrane leak (provided a membrane was used) .
can you post some pics?
Caulk and grout are NOT considered water proof things. They are used to limit the "loading" of a sound design.
Water leaks are a pain even under the best of circumstances..
The leak could be from a number of sources in the bathroom; leaking supply lines, leaking drain lines, leaking shower, leaking water closet (toilet) seal...and to make things more complicated, it doesn't even have to be bathroom related....
Once between the floors, water can run several feet, depending on the joist layout and other factors. What may appear to be a leak under a specific fixture, may in fact be from a nearby source. There are a number of factors that can change when the leak is actually detected on a ceiling as well...It is easy to jump on cracking grout in a shower pan, when it may be something else.
Did anyone do anything to the shower before the leak appeared? I assume the plumber came after the leak?
Custom built shower pans installed by professionals are built in such a way that they expect moisture to get behind/under the tile and as such are designed to funnel this moisture safely into a proper shower drain fixture. A rubber membrane on the pan and a waterproof membrane behind the tile are two of the components of this system designed to handle the water in a shower. As Kevin (KMS) alluded to, properly sealed tile and grout will limit and sometimes eliminate at times the job this waterproof system that you can't see has to do.
Given enough time and enough patience, there are ways to narrow down what is leaking...but not necessarily pinpoint it. If the caulk did stop the leak, it is important to understand that fixed the symptom by plugging the hole upstream.....but it didn't necessarily fix the real problem...in that you have a leak in the dam downstream (membrane/fitting leak). The next time that grout lets more water through for whatever reason, it will more than likely leak again.
In a similar situation, I would troubleshoot as best I could to see if we could narrow it down. It may mean cutting out some of the kitchen ceiling so they can get a good look at it.
This link shows some pics of the process.