diff --git a/ext/PigeonsJuliaBUGSExt/interface.jl b/ext/PigeonsJuliaBUGSExt/interface.jl
index d44efe5b5..2b77927d3 100644
--- a/ext/PigeonsJuliaBUGSExt/interface.jl
+++ b/ext/PigeonsJuliaBUGSExt/interface.jl
@@ -80,5 +80,18 @@ function Pigeons.sample_iid!(log_potential::JuliaBUGSLogPotential, replica, shar
     replica.state = _sample_iid(log_potential.private_model, replica.rng)
 end
 
+# parameter names
 Pigeons.sample_names(::Vector, log_potential::JuliaBUGSLogPotential) = 
     [(Symbol(string(vn)) for vn in log_potential.private_model.parameters)...,:log_density]
+
+# Parallelism invariance
+Pigeons.recursive_equal(a::Union{JuliaBUGSPath,JuliaBUGSLogPotential}, b) =
+    Pigeons._recursive_equal(a,b)
+function Pigeons.recursive_equal(a::T, b) where T <: JuliaBUGS.BUGSModel
+    included = (:transformed, :model_def, :data)
+    excluded = Tuple(setdiff(fieldnames(T), included))
+    Pigeons._recursive_equal(a,b,excluded)
+end
+# just check the betas match, the model is already checked within path
+Pigeons.recursive_equal(a::AbstractVector{<:JuliaBUGSLogPotential}, b) =    
+    all(lp1.beta == lp2.beta for (lp1,lp2) in zip(a,b))
diff --git a/test/test_JuliaBUGS.jl b/test/test_JuliaBUGS.jl
index 36263f48f..b0fad3a6e 100644
--- a/test/test_JuliaBUGS.jl
+++ b/test/test_JuliaBUGS.jl
@@ -63,14 +63,14 @@ end
     @test res.passed
 end
 
-@testset "Model with mixed state types using MPI" begin
+@testset "Parallelism invariance using MPI" begin
     target=incomplete_count_data()
     r = pigeons(;
-        target,
+        target=unid_target,
         n_rounds = 5,
         n_chains = 4,
         checkpoint = true,
-        # checked_round = 4, # NB: doesn't work yet, need a fine-tuned equality check for JuliaBUGS.BUGSModel
+        checked_round = 4,
         multithreaded = true,
         on = ChildProcess(
             n_local_mpi_processes = set_n_mpis_to_one_on_windows(2),