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)

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)




 
[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/ojzBA.png