Is it theoretically possible to have a universe where sound travels faster than light $c$? We all "know" nothing can travel faster than light. However, if we're allowed to tweak the fundamental constants of nature, is it theoretically possible that such an universe might exist? 
Update:
I refer to the speed of light and sound in a colloquial way (maybe I shouldn't have, since this is a Physics site). Example: in a thunderstorm, we'd hear the thunderclap before we see the lightning. (I assume that light remains massless, and that the index of refractive $n=1$ of the medium is one.)
 A: We have to be careful about what we mean by "speed of light". It can mean two things: the speed at which light travels, which I'll write as $s_{light}$, and the maximum speed at which anything can possibly travel, which is written $c$. In our universe, in a vacuum, $s_{light}=c$, as far as we know.
Now, no information can ever be transmitted faster than $c$, unless we're willing to tweak the universe so much that it no longer obeys relativity (in which case we probably shouldn't be posting on a physics site). Information can be transmitted by sound, so there's no way sound could ever possibly travel faster than $c$.
However, it is possible for light to travel slower than $c$. Even in the real universe you can have $s_{light}<c$ if you're not in a vacuum. If photons had mass they would even travel slower than $c$ in a vacuum. (Though they wouldn't really be photons anymore and we're edging towards being off-topic again). In such cases I don't believe there's any special reason why $v_{light}$ would have to be faster than the speed of sound. It's possible in principle in our universe, although there certainly isn't any material in which it actually happens, at least under non-extreme temperatures and pressures.
A: Sound is a pressure wave, and the generation of a pressure gradient requires atoms/molecules to move to create a density difference. No particle can move faster than light, so it's impossible to create a pressure gradient that propagates faster than light.
Nathaniel's argument that sound waves could travel faster than light in a system like a Bose Einstein condensate is ingenious and true in the sense that light can be brought to a virtual standstill in a BEC. However this is cheating a bit as light isn't just light in those conditions. The light and the BEC form a composite system called a polariton and it's the polariton moving slower than $c$ not light. I certainly agree with Nathaniel that sound can move faster than a polariton under the right conditions.
A: There is a "universe" where the speed of sound is greater than the speed of light (or at least the speed of electromagnetic wave propagation), and that is inside conductors.
In a conductor, the EM wave velocity is
$$ v = c \left(\frac{2 \omega \epsilon_0}{\sigma}\right)^{1/2},$$
where $\sigma$ is the conductivity and $\omega$ is the angular wave frequency.
Putting some numbers in for copper at 10Mhz, I get a "light" speed of $\sim 1000$ m/s, whereas the sound speed in copper is about 5000 m/s.
Yes, I know that's not what the OP meant, this is not the same as having a much lower value for $c$. But interesting nonetheless.
A: 
We all "know" nothing can travel faster than light [...]

When referring to "the speed of light (in vacuum)" in the context of relativity we know definitely that we mean "signal front speed"; and the notions of "signal" and "signal front" are considered definite and unambiguous as well.

[...] a universe where sound travels faster than light?

By "sound travelling" surely we don't mean "just independent, meaningless noise", but some specific ("acoustic") sort of transmitting a signal.
Consequently, the "speed of sound" which refers to this specific ("acoustic") sort of transmitting a signal must definitely be less, or at most equal, to the signal front speed which refers to the (all-inclusive) corresponding signal front.

if we're allowed to tweak the fundamental constants of nature

Note that $c_0$, i.e. the signal front speed, is a purely formal non-zero constant; independent of any particular choice of units, such as for instance the choice of SI-units "m" and "s".
So there is nothing to be "tweaked" with this constant, as a matter of principle, aside of the mere formality of choosing some particular symbol to denote it.
Remark related to the recent OP Update: 

I refer to the speed of light and sound in a colloquial way [...] 

There is of course no way to answer a question about which colloquialisms might be used in alternate universes for denoting and expressing any underlying definite and unambiguous physical, geometrical, and/or causality-related notions.
Also, I've thought about how "the OP question could be improved", or which expression of what I believe is the main point of the present question I consider generally acceptable; and I'd phrase it so:
 "How to determine the credibility of a possible claim of "super-luminal signalling" ? (PSE/q/145226)".
