Não creio que exista uma maneira “fácil” de alcançar esse resultado; quando você reordena seu eixo y de acordo com max(upper)a ordem 'definida' para todos os domínios (embora eu ficaria feliz em provar que estou errado nisso).

Uma possível solução alternativa seria dividir seus gráficos por classe e reordenar cada um separadamente, por exemplo

library(tidyverse)

mydat <- data.frame(mean = c(23, 24, 15, 27, 18, 19, 23, 20, 32),
                    lower = c(20, 19, 13, 15, 14, 18, 20, 17, 20),
                    upper = c(25, 29, 17, 39, 22, 20, 26, 23, 40),
                    class = c("A", "B", "C", "A", "B", "C", "A", "B", "C"),
                    domain = c("North", "North", "North", "West", "West", "West", "South", "South", "South"))
mydat$class <- as.factor(mydat$class)
mydat$domain <- as.factor(mydat$domain)

colour_palette <- scales::hue_pal()(3)

mydat %>%
  group_split(class) %>%
  map(~{ggplot(data = .x,
               aes(x = mean, y = fct_reorder(.f = class, .x = upper, .fun = max),
                   color = fct_reorder(.f = domain, .x = upper, .fun = max),
                   Group = fct_reorder(.f = domain, .x = upper, .fun = max),
                   xmin = lower, xmax = upper)) +
  geom_errorbar(width = 0.1, position = position_dodge(width = 0.5)) + 
  theme_light() + 
  geom_point(position = position_dodge(width = 0.5)) +
      xlab("Counts") +
      ylab("Class") +
      scale_color_manual(values = c("North" = colour_palette[1],
                                    "South" = colour_palette[2],
                                    "West" = colour_palette[3]),
                         name = "Domain")})
#> [[1]]

#> 
#> [[2]]

#> 
#> [[3]]

^{Criado em 23/10/2023 com reprex v2.0.2}

Se você fizer isso 'manualmente', poderá organizá-los na ordem desejada, por exemplo

library(tidyverse)

mydat <- data.frame(mean = c(23, 24, 15, 27, 18, 19, 23, 20, 32),
                    lower = c(20, 19, 13, 15, 14, 18, 20, 17, 20),
                    upper = c(25, 29, 17, 39, 22, 20, 26, 23, 40),
                    class = c("A", "B", "C", "A", "B", "C", "A", "B", "C"),
                    domain = c("North", "North", "North", "West", "West", "West", "South", "South", "South"))
mydat$class <- as.factor(mydat$class)
mydat$domain <- as.factor(mydat$domain)

colour_palette <- scales::hue_pal()(3)

p1 <- mydat %>%
  filter(class == "A") %>%
  ggplot(aes(x = mean, y = fct_reorder(.f = class, .x = upper, .fun = max),
                   color = fct_reorder(.f = domain, .x = upper, .fun = max),
                   Group = fct_reorder(.f = domain, .x = upper, .fun = max),
                   xmin = lower, xmax = upper)) +
      geom_errorbar(width = 0.1, position = position_dodge(width = 0.5)) + 
      theme_light(base_size = 18) + 
      geom_point(position = position_dodge(width = 0.5)) +
      xlab("") +
      ylab("") +
      scale_color_manual(values = c("North" = colour_palette[1],
                                    "South" = colour_palette[2],
                                    "West" = colour_palette[3]),
                         name = "Domain") +
  scale_x_continuous(limits = c(10, 45))

p2 <- mydat %>%
  filter(class == "B") %>%
  ggplot(aes(x = mean, y = fct_reorder(.f = class, .x = upper, .fun = max),
             color = fct_reorder(.f = domain, .x = upper, .fun = max),
             Group = fct_reorder(.f = domain, .x = upper, .fun = max),
             xmin = lower, xmax = upper)) +
  geom_errorbar(width = 0.1, position = position_dodge(width = 0.5)) + 
  theme_light(base_size = 18) + 
  geom_point(position = position_dodge(width = 0.5)) +
  xlab("Counts") +
  ylab("") +
  scale_color_manual(values = c("North" = colour_palette[1],
                                "South" = colour_palette[2],
                                "West" = colour_palette[3]),
                     name = "Domain") +
  theme(legend.position = "none") +
  scale_x_continuous(limits = c(10, 45))

p3 <- mydat %>%
  filter(class == "C") %>%
  ggplot(aes(x = mean, y = fct_reorder(.f = class, .x = upper, .fun = max),
             color = fct_reorder(.f = domain, .x = upper, .fun = max),
             Group = fct_reorder(.f = domain, .x = upper, .fun = max),
             xmin = lower, xmax = upper)) +
  geom_errorbar(width = 0.1, position = position_dodge(width = 0.5)) + 
  theme_light(base_size = 18) + 
  geom_point(position = position_dodge(width = 0.5)) +
  xlab("") +
  ylab("") +
  scale_color_manual(values = c("North" = colour_palette[1],
                                "South" = colour_palette[2],
                                "West" = colour_palette[3]),
                     name = "Domain") +
  theme(legend.position = "none") +
  scale_x_continuous(limits = c(10, 45))

library(patchwork)
p3 / p1 / p2

^{Criado em 23/10/2023 com reprex v2.0.2}

Não consigo descobrir o mesmo resultado que você esperava, mas espero que o código a seguir faça sentido para você de forma equivalente (onde domainas informações são adicionadas como rótulos a cada barra de erros)

mydat %>%
    mutate(class = factor(class, names(sort(by(upper, class, max))))) %>%
    arrange(class, desc(upper)) %>%
    mutate(color = factor(n() - row_number()), .by = class) %>%
    ggplot(aes(x = mean, y = class, color = color, Group = domain, xmin = lower, xmax = upper)) +
    geom_errorbar(width = 0.1, position = position_dodge(width = 0.5)) +
    geom_text(aes(label = domain), position = position_dodge(0.5)) +
    theme_light() +
    geom_point(position = position_dodge(width = 0.5)) +
    xlab("Counts")

Dados

mydat <- data.frame(
    mean = c(23, 24, 15, 27, 18, 19, 23, 20, 32),
    lower = c(20, 19, 13, 15, 14, 18, 20, 17, 20),
    upper = c(25, 29, 17, 39, 22, 20, 26, 23, 40),
    class = c("A", "B", "C", "A", "B", "C", "A", "B", "C"),
    domain = c("North", "North", "North", "West", "West", "West", "South", "South", "South")
)