diff --git a/common/pseudorand.c b/common/pseudorand.c
index 7f3efafe5..52da1e61e 100644
--- a/common/pseudorand.c
+++ b/common/pseudorand.c
@@ -26,25 +26,42 @@ static void init_if_needed(void)
 	}
 }
 
-uint64_t pseudorand(uint64_t max)
+uint64_t pseudorand_(uint64_t max, uint64_t *offset)
 {
 	init_if_needed();
 
 	assert(max);
+
+	/* We try to avoid being order-dependent here. */
+	if (randbytes_overridden()) {
+		uint64_t rand;
+		randbytes_(&rand, sizeof(rand), offset);
+		return rand % max;
+	}
 	return isaac64_next_uint(&isaac64, max);
 }
 
-uint64_t pseudorand_u64(void)
+uint64_t pseudorand_u64_(uint64_t *offset)
 {
 	init_if_needed();
 
+	if (randbytes_overridden()) {
+		uint64_t rand;
+		randbytes_(&rand, sizeof(rand), offset);
+		return rand;
+	}
 	return isaac64_next_uint64(&isaac64);
 }
 
-double pseudorand_double(void)
+double pseudorand_double_(uint64_t *offset)
 {
 	init_if_needed();
 
+	if (randbytes_overridden()) {
+		uint64_t rand;
+		randbytes_(&rand, sizeof(rand), offset);
+		return rand / (double)UINT64_MAX;
+	}
 	return isaac64_next_double(&isaac64);
 }
 
diff --git a/common/pseudorand.h b/common/pseudorand.h
index ebe9e0b59..0e5ce6c56 100644
--- a/common/pseudorand.h
+++ b/common/pseudorand.h
@@ -8,18 +8,21 @@
 /**
  * pseudorand - pseudo (guessable!) random number between 0 and max-1.
  */
-uint64_t pseudorand(uint64_t max);
+#define pseudorand(max) ({static uint64_t offset; pseudorand_((max), &offset);})
+uint64_t pseudorand_(uint64_t max, uint64_t *offset);
 
 /**
- * pseudorand - pseudo (guessable!) random number between 0 and UINT64_MAX.
+ * pseudorand_u64 - pseudo (guessable!) random number between 0 and UINT64_MAX.
  */
-uint64_t pseudorand_u64(void);
+#define pseudorand_u64() ({static uint64_t offset; pseudorand_u64_(&offset);})
+uint64_t pseudorand_u64_(uint64_t *offset);
 
 /**
  * pseudorand - pseudo (guessable!) random number between 0 (inclusive) and 1
  * (exclusive).
  */
-double pseudorand_double(void);
+#define pseudorand_double()  ({static uint64_t offset; pseudorand_double_(&offset);})
+double pseudorand_double_(uint64_t *offset);
 
 /**
  * Get the siphash seed for hash tables.
diff --git a/common/test/run-psbt_diff.c b/common/test/run-psbt_diff.c
index 235163fe0..1c107851a 100644
--- a/common/test/run-psbt_diff.c
+++ b/common/test/run-psbt_diff.c
@@ -38,9 +38,9 @@ u8 fromwire_u8(const u8 **cursor UNNEEDED, size_t *max UNNEEDED)
 /* Generated stub for fromwire_u8_array */
 void fromwire_u8_array(const u8 **cursor UNNEEDED, size_t *max UNNEEDED, u8 *arr UNNEEDED, size_t num UNNEEDED)
 { fprintf(stderr, "fromwire_u8_array called!\n"); abort(); }
-/* Generated stub for pseudorand_u64 */
-uint64_t pseudorand_u64(void)
-{ fprintf(stderr, "pseudorand_u64 called!\n"); abort(); }
+/* Generated stub for pseudorand_u64_ */
+uint64_t pseudorand_u64_(uint64_t *offset UNNEEDED)
+{ fprintf(stderr, "pseudorand_u64_ called!\n"); abort(); }
 /* Generated stub for towire */
 void towire(u8 **pptr UNNEEDED, const void *data UNNEEDED, size_t len UNNEEDED)
 { fprintf(stderr, "towire called!\n"); abort(); }
diff --git a/onchaind/test/run-grind_feerate-bug.c b/onchaind/test/run-grind_feerate-bug.c
index 4e64c268b..e57aca2a4 100644
--- a/onchaind/test/run-grind_feerate-bug.c
+++ b/onchaind/test/run-grind_feerate-bug.c
@@ -175,6 +175,9 @@ void peer_billboard(bool perm UNNEEDED, const char *fmt UNNEEDED, ...)
 /* Generated stub for randbytes_ */
 void randbytes_(void *bytes UNNEEDED, size_t num_bytes UNNEEDED, u64 *offset UNNEEDED)
 { fprintf(stderr, "randbytes_ called!\n"); abort(); }
+/* Generated stub for randbytes_overridden */
+bool randbytes_overridden(void)
+{ fprintf(stderr, "randbytes_overridden called!\n"); abort(); }
 /* Generated stub for shachain_get_secret */
 bool shachain_get_secret(const struct shachain *shachain UNNEEDED,
 			 u64 commit_num UNNEEDED,
diff --git a/tests/fuzz/libfuzz.c b/tests/fuzz/libfuzz.c
index aada6f8ce..00e4b84d0 100644
--- a/tests/fuzz/libfuzz.c
+++ b/tests/fuzz/libfuzz.c
@@ -21,7 +21,7 @@ int LLVMFuzzerInitialize(int *argc, char ***argv);
 /* Provide a non-random pseudo-random function to speed fuzzing. */
 static isaac64_ctx isaac64;
 
-uint64_t pseudorand(uint64_t max)
+uint64_t pseudorand_(uint64_t max, uint64_t *offset)
 {
 	assert(max);
 	return isaac64_next_uint(&isaac64, max);
diff --git a/wire/test/run-peer-wire.c b/wire/test/run-peer-wire.c
index a220af8fc..cc6886456 100644
--- a/wire/test/run-peer-wire.c
+++ b/wire/test/run-peer-wire.c
@@ -25,6 +25,9 @@ extern secp256k1_context *secp256k1_ctx;
 /* Generated stub for randbytes_ */
 void randbytes_(void *bytes UNNEEDED, size_t num_bytes UNNEEDED, u64 *offset UNNEEDED)
 { fprintf(stderr, "randbytes_ called!\n"); abort(); }
+/* Generated stub for randbytes_overridden */
+bool randbytes_overridden(void)
+{ fprintf(stderr, "randbytes_overridden called!\n"); abort(); }
 /* AUTOGENERATED MOCKS END */
 
 /* memsetting pubkeys doesn't work */
diff --git a/wire/test/run-tlvstream.c b/wire/test/run-tlvstream.c
index 3782191da..5aa609336 100644
--- a/wire/test/run-tlvstream.c
+++ b/wire/test/run-tlvstream.c
@@ -33,6 +33,9 @@ bool fromwire_channel_id(const u8 **cursor UNNEEDED, size_t *max UNNEEDED,
 /* Generated stub for randbytes_ */
 void randbytes_(void *bytes UNNEEDED, size_t num_bytes UNNEEDED, u64 *offset UNNEEDED)
 { fprintf(stderr, "randbytes_ called!\n"); abort(); }
+/* Generated stub for randbytes_overridden */
+bool randbytes_overridden(void)
+{ fprintf(stderr, "randbytes_overridden called!\n"); abort(); }
 /* Generated stub for towire_channel_id */
 void towire_channel_id(u8 **pptr UNNEEDED, const struct channel_id *channel_id UNNEEDED)
 { fprintf(stderr, "towire_channel_id called!\n"); abort(); }