diff --git a/stdlib/Distributed/src/clusterserialize.jl b/stdlib/Distributed/src/clusterserialize.jl
index 6bca816687af3..c50393caeb0f5 100644
--- a/stdlib/Distributed/src/clusterserialize.jl
+++ b/stdlib/Distributed/src/clusterserialize.jl
@@ -2,7 +2,7 @@
 
 using Serialization: serialize_cycle, deserialize_cycle, writetag,
                      serialize_typename, deserialize_typename,
-                     TYPENAME_TAG, reset_state, serialize_type
+                     TYPENAME_TAG, TASK_TAG, reset_state, serialize_type
 using Serialization.__deserialized_types__
 
 import Serialization: object_number, lookup_object_number, remember_object
@@ -102,6 +102,26 @@ function serialize(s::ClusterSerializer, t::Core.TypeName)
     nothing
 end
 
+function serialize(s::ClusterSerializer, t::Task)
+    serialize_cycle(s, t) && return
+    if istaskstarted(t) && !istaskdone(t)
+        error("cannot serialize a running Task")
+    end
+    writetag(s.io, TASK_TAG)
+    serialize(s, t.code)
+    serialize(s, t.storage)
+    bt = t.backtrace
+    if bt !== nothing
+        if !isa(bt, Vector{Any})
+            bt = Base.process_backtrace(bt, 100)
+        end
+        serialize(s, bt)
+    end
+    serialize(s, t.state)
+    serialize(s, t.result)
+    serialize(s, t.exception)
+end
+
 function serialize(s::ClusterSerializer, g::GlobalRef)
     # Record if required and then invoke the default GlobalRef serializer.
     sym = g.name
@@ -231,6 +251,23 @@ function deserialize(s::ClusterSerializer, t::Type{<:CapturedException})
     return CapturedException(capex, bt)
 end
 
+function deserialize(s::ClusterSerializer, ::Type{Task})
+    t = Task(nothing)
+    deserialize_cycle(s, t)
+    t.code = deserialize(s)
+    t.storage = deserialize(s)
+    state_or_bt = deserialize(s)
+    if state_or_bt isa Symbol
+        t.state = state_or_bt
+    else
+        t.backtrace = state_or_bt
+        t.state = deserialize(s)
+    end
+    t.result = deserialize(s)
+    t.exception = deserialize(s)
+    t
+end
+
 """
     clear!(syms, pids=workers(); mod=Main)
 
diff --git a/stdlib/Distributed/test/distributed_exec.jl b/stdlib/Distributed/test/distributed_exec.jl
index 379a2b5b46fb5..6bc3cef045fd6 100644
--- a/stdlib/Distributed/test/distributed_exec.jl
+++ b/stdlib/Distributed/test/distributed_exec.jl
@@ -1682,6 +1682,19 @@ let (h, t) = Distributed.head_and_tail(Int[], 0)
     @test collect(t) == []
 end
 
+# issue #35937
+let e
+    try
+        pmap(1) do _
+            wait(@async error(42))
+        end
+    catch ex
+        e = ex
+    end
+    # check that the inner TaskFailedException is correctly formed & can be printed
+    @test sprint(showerror, e) isa String
+end
+
 include("splitrange.jl")
 
 # Run topology tests last after removing all workers, since a given