Barnard,[4][5] the test did not gain popularity due to the computational difficulty of calculating the p value and Fisher’s specious disapproval.
To distinguish the different types of designs, suppose a researcher is interested in testing whether a treatment quickly heals an infection.
Fisher's exact test avoids estimating the nuisance parameter(s) by conditioning on both margins, an approximately ancillary statistic that constrains the possible outcomes.
The problem with that Fisher's procedure is that it excludes some of the outcomes which are possiblities when there is no constraint on the total numbers in each column and row.
Barnard's test can be applied to larger tables, but the computation time increases and the power advantage quickly decreases.