yeah, I'm with BuddhUU...
I can't imagine a Fluke with intonation problems... especially if it happens to be the plastic fret board.
so much so that I would have to assume some other things must be in play.
i would probably recommend a string change also, but before that, I would double check a few very basic things...
please forgive me if my suggestions sound too basic. I am not trying to insult anyone's intelligence, but since I don't know your ukulele level, I am working under the assumption of "none"...
(edit: when I first wrote the below tuning suggestions I missed the part in you post where you mention your chromatic tuner. and the other advice is still okay for a full on newbie, but after rereading your post, I have concluded that you are not. enjoy the information anyways )
first, although I would prefer a more accurate tuning strategy than a pitch pipe... say, a clip on chromatic tuner maybe?But, pitch pipe is okay too. I would double check that the pitch pipe is still good. use a tuned other instrument like a piano or electric keyboard or, better yet, go to one of the online tuning resources, like here...
http://www.get-tuned.com/ukulele_tuner.php
either double check your pitch pipe and/or tune you uke directly.
okay. there are a couple other basic things... the pitch pipe is most likely tuned to the notes G, C, E, and A... first make sure that that is the case. if it is, then you can try to get your uke strings to match those notes... depending on how you hold your uke (lefty? righty?) the top/upper/highest string when the length of the uke is horizontal should be your G string... the next one down, the C, then the E, and the bottom/lowest string should be A. if not make them so. and the tension on the strings should be roughly similar... if not the strings were probably strung for a "handedness" (left handed vs. right handed) opposite from you.
now, this next thing may not solve your discordant sound issues, but might as well cover all the bases... once you got your strings in the right order for your "handedness" and they are in tune, make sure that that "C" chord you are trying to hold is a "C" chord for GCEA tuning. If you are using a chord position chart for some other tuning, it will sound like a chord but it will not be the chord that is named. for example, if you are trying to hold a C chord using a Baritone uke (DGBE tuning) chart it will most likely be noted as 2010 (hold down top string second fret, next string open, next string first fret, and bottom string open)... on a GCEA tuned uke that chord is an F chord. That said, that C chord you are trying (for GCEA tuning) is probably theone noted as 0003 (the bottom string, the A string, is held at the third fret)
okay, with all that out of the way, let's also double check your finger positioning on the fretboard. make sure when you hold a string down at a particular fret, that your finger is not directly on top of the actual raised fret. the finger should be as close to the fret as possible but in the space just before the fret (the space towards the tuners pegs not the space towards the sound hole). touching the fret on the side is okay but if you hold directly on the fret there is a good chance you will be off.
that's most of the basic stuff i can think of at the moment. if that doesn't work, I would attempt the recommended string change. Any of the higher end strings will suffice IMO. I usually just tell people to avoid GHS strings... but if your current problem turns out to be a string issue only, it is likely because your current set is defective and, really, any new set of decent uke strings (even GHS) should fix it.
good luck!