Negative modulus give rise to acoustic metamaterials. 

https://en.wikipedia.org/wiki/Acoustic_metamaterial

Thank you, Vikash, but are you sure this is the case? Acoustic metamaterials are defined by a negative bulk modulus - I'm talking about a negative shear loss modulus (the imaginary component of the shear modulus). 

Or am I missing something?

See this, http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.107.105506

Negative loss would mean the specimen was putting power into the oscillation.  If I really exercise my imagination, I can come up with ways this could happen, but not without going to a lot of trouble to create an external drive mechanism...which is clearly not your case. 

The fact that the negative loss appeared at 500 Hz may be significant to an explanation. In Rheometrics (TM) equipment this used to be the maximum frequency, though I'm not necessarily up to date now.  If odd behavior caused by instrument error is going to show up, I would expect it most likely at one extreme or the other of the instrument's range.

Can you get at the same region but at a lower frequency by temperature shift? 

The DMA measurements went up to 1000Hz, but I'm still highly skeptical about their veracity. I plan to recalibrate the equipment this week, and repeat the measurements before proceeding to any conclusions. If it is not a systematic error, I will try to follow your suggestions. Thank you so much

Still thinking about how the instrument could do this. If it errs enough in the phase angle between driver displacement and driven force, you could seen negative loss modulus.

Determination of the phase requires that the frame of the instrument be much stiffer than what you measure. It also requires that no subcomponents of the instrument frame have significant resonances that couple to the measuring zone. Basically, if some part of the instrument is flopping or shaking, it could under some circumstances make the apparent measured phase lead rather than lag. 

I recall hearing Stanley S. Sternstein tell of all the grief he was having building his own mechanical spectrometer to operate above 1000 Hz, as an aside he made to our class about 35 years ago.  He described these stiffness and coupled-oscillator issues.

When you do your calibration, you may wish to choose a steel shim of about the same total stiffness as your experimental specimen.  You might also wish to check whether you continue to get apparent signal when you have no specimen coupling between driving tool and sensor, just a big air gap.  Sorry if these are obvious. 

Not obvious at all! As I said, I'm new to rheology and DMA, so every tidbit of advice is most welcomed. Thank you again