Maybe I can start by showing you a permutation test for the ratio of the geometric means of two samples of size 20 from two different gamma populations (which, of course, have positive support).
set.seed(2021)
x1 = rgamma(20, 5, .10)
x2 = rgamma(20, 5, .2)
boxplot(x1, x2, col="skyblue2", horizontal=T)
The respective geometric means are $42.06$ and $24.70.$ Their ratio
is rto.obs, $1.703.$
Because it seems natural to look at ratios of geometric means,
the question is whether $1.703$ is significantly different from $1$
(2-sided test) at the 5% level of significance.
g1 = prod(x1)^.05; g2 = prod(x2)^.05
g1; g2
[1] 42.06336
[1] 24.69576
rto.obs = g1/g2; rto.obs
[1] 1.703262
Possibly, one could derive the the distribution of the ratio of gamma geometric means to find an exact test. Instead, I will repeatedly scramble the $n_1+n_2 = 40$ observations into two permuted samples each with $n_i = 20$ observations, find the two geometric means, and then their ratio. After doing this $B = 10\,000$ times, I will have $B$ ratios, which can give me a good idea of the distribution of the ratio. Finally, comparing this simulated permutation distribution of ratios with the observed ratio $1.703,$ I can see whether or not the observed ratio is unusually far from $1.$
set.seed(602)
x = c(x1, x2)
B = 10000; rto.prm = numeric(B)
for(i in 1:B) {
x.prm = sample(x)
rto.prm[i] = prod(x.prm[1:20])^.05/prod(x.prm[21:40])^.05
}
mean(rto.prm/rto.obs > .975 | rto.obs/rto.prm < .025)
[1] 0.0047 # P-value of sim permutation test.
hdr = "Simulated Dist'n of Ratios of Geometric Means"
hist(rto.prm, prob=T, col="skyblue2", main=hdr)
abline(v=rto.obs);abline(v=1/rto.obs)
