Consider the following example, a negative binomial model with an offset fit by glmmTMB:

# 1. Load the required package
library(glmmTMB)

# 2. Prepare the built-in Salamanders dataset
data("Salamanders")

# For reproducibility, set a seed
set.seed(42)

# Create a mock "area_searched" variable representing plot size (e.g., between 10 and 100 sq meters)
Salamanders$area_searched <- runif(nrow(Salamanders), min = 10, max = 100)

# 3. Fit a zero-inflated negative binomial model with an offset
# We use log(area_searched) to evaluate the count as a rate (salamanders per square meter)
fit_offset <- glmmTMB(
  count ~ mined + cover + offset(log(area_searched)) + (1 | site),
  ziformula = ~ mined,
  family = nbinom2,
  data = Salamanders
)

# 4. View the model summary
summary(fit_offset)

This results in

 Family: nbinom2  ( log )
Formula:          count ~ mined + cover + offset(log(area_searched)) + (1 | site)
Zero inflation:         ~mined
Data: Salamanders

      AIC       BIC    logLik -2*log(L)  df.resid 
   1820.0    1851.3    -903.0    1806.0       637 

Random effects:

Conditional model:
 Groups Name        Variance Std.Dev.
 site   (Intercept) 0.3158   0.562   
Number of obs: 644, groups:  site, 23

Dispersion parameter for nbinom2 family (): 0.533 

Conditional model:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -4.60591    0.39227 -11.742  < 2e-16 ***
minedno      1.67837    0.46724   3.592 0.000328 ***
cover       -0.05222    0.18562  -0.281 0.778463    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Zero-inflation model:
              Estimate Std. Error z value Pr(>|z|)
(Intercept)   -0.06702    0.44813  -0.150    0.881
minedno      -18.22190 3484.00726  -0.005    0.996

so far so good.

But If I now replace offset in the formula with stats::offset, a real regressor (i.e. coefficient not fixed to 1) is inserted:

# 3. Fit a zero-inflated negative binomial model with an offset
# We use log(area_searched) to evaluate the count as a rate (salamanders per square meter)
fit_offset <- glmmTMB(
  count ~ mined + cover + stats::offset(log(area_searched)) + (1 | site),
  ziformula = ~ mined,
  family = nbinom2,
  data = Salamanders
)

# 4. View the model summary
summary(fit_offset)

which results in:

 Family: nbinom2  ( log )
Formula:          count ~ mined + cover + stats::offset(log(area_searched)) + (1 |      site)
Zero inflation:         ~mined
Data: Salamanders

      AIC       BIC    logLik -2*log(L)  df.resid 
   1743.3    1779.0    -863.6    1727.3       636 

Random effects:

Conditional model:
 Groups Name        Variance Std.Dev.
 site   (Intercept) 0.03921  0.198   
Number of obs: 644, groups:  site, 23

Dispersion parameter for nbinom2 family (): 1.05 

Conditional model:
                                  Estimate Std. Error z value Pr(>|z|)    
(Intercept)                       -0.12946    0.62084  -0.209 0.834815    
minedno                            1.41575    0.38738   3.655 0.000257 ***
cover                             -0.18904    0.11078  -1.706 0.087928 .  
stats::offset(log(area_searched)) -0.06322    0.11289  -0.560 0.575491    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Zero-inflation model:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)   0.5627     0.5122   1.099 0.271930    
minedno      -2.3698     0.6099  -3.886 0.000102 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Why is that?