Tweedie distribution without zeroes Recently I've found the Tweedie distribution useful for modelling my plant shoot weight data in glmmTMB. I started using it because my shoot weight datasets have many zeroes, and it has yielded better fits than I was able to get with zero-inflated models. 
One of my shoot weight datasets doesn't have zeroes, however (although it does have many <1 values). The Gaussian distribution fits this dataset well enough, but I don't think that the predictions it makes are correct. I tried this model with the Tweedie distribution, and I got a better fit and predictions that make more sense.
Despite the good fit, is it invalid to use the Tweedie distribution when your data doesn't have zeroes? This is the code I'm using:
Model4 <- glmmTMB(Shoot.weight ~ Species + N.Level + Rhizobia + Species:N.Level + 
                      Species:Rhizobia + N.Level:Rhizobia + N.Level:Rhizobia:Species + 
                     (1 | Block), family = tweedie, data = SOY)

I believe that the specific Tweedie distribution that I'm using here is the compound Poisson-Gamma distribution, but correct me if I'm wrong about that.
 A: Tweedie distributions don't always have zeros, but even when they do, often the proportion of zeros in fitted Tweedie models is often fairly small - at least in the cases I've seen it used to fit.  (Your post indicates you have $1<p<2$, so that would imply some zeroes in the model)
I think the Tweedie family is of relatively modest value for modelling data with zeroes: the proportion of zeroes you get in the model is not only affected by the proportion of zeroes in the data -- it's a function of $p$, $\mu$ and $\sigma^2$ and a GLM will work with a fixed $p$, while the estimates of $\mu$ and $\sigma^2$ will be affected by all the data. 
However, the family can be quite handy if you want GLMs with variance functions that are a power of the mean.
You could attempt to identify the proportion of zeroes you're actually getting in your fitted model. I think that the proportion of zeroes is $\exp(-\frac{\mu^{2-p}}{(2-p)\sigma^2})$ (i.e. it varies with the fitted mean).  
I wouldn't worry terribly much whether your data have zeros unless your fitted model suggests a fairly substantial proportion of them somewhere that you have a fair bit of data.
