Slightly different to ggplot2, the ternary axis and labels are presented on EVERY PANEL, so therefore, if facet_wrap or facet_grid is used, the axis labels and apex titles are present on each panel. This can be demonstrated quite easily, consider the following example:

set.seed(1) #for reproduceability
MYDATA <- data.frame(A=runif(100),
                     B=runif(100),
                     C=runif(100),
                     groupA=rep(paste(c("A1","A2"),""),50)[sample(50)],
                     groupB=rep(paste(c("B1","B2"),""),50)[sample(50)]) 

#create plot
plot <- ggtern(data=MYDATA,mapping=aes(x=A,y=B,z=C)) + 
        labs(title ="Example Ternary Plots w/ Facetting",
             fill  ="Fill Metric",
             color ="Color Metric",
             alpha ="Alpha Metric",
             x     ="T",
             y     ="L",
             z     ="R") + 
        facet_grid(groupA~groupB) +
        stat_density2d(fullrange=T,n=200,
                       geom="polygon",
                       aes(fill  =..level..,
                           alpha =..level..)) + 
        theme_tern_rgbw()  + #custom themes 
        atomic_percent()   + #make ternary scales on atomic %
        geom_point(size=3,shape=16,
                   aes(colour=factor(paste0(groupA,"/",groupB))))
#render
plot

Which yields the following pleasing result:

Faceting Example

Advanced Faceting Example