diff --git a/dtool/src/prc/streamWrapper.I b/dtool/src/prc/streamWrapper.I
index 0f9d65d414..6d2578262c 100644
--- a/dtool/src/prc/streamWrapper.I
+++ b/dtool/src/prc/streamWrapper.I
@@ -58,6 +58,24 @@ release() {
   _lock.unlock();
 }
 
+/**
+ * Increments the reference count.  Only has impact if the class that manages
+ * this StreamWrapper's lifetime (eg. Multifile) respects it.
+ */
+INLINE void StreamWrapperBase::
+ref() const {
+  AtomicAdjust::inc(_ref_count);
+}
+
+/**
+ * Decrements the reference count.  Only has impact if the class that manages
+ * this StreamWrapper's lifetime (eg. Multifile) respects it.
+ */
+INLINE bool StreamWrapperBase::
+unref() const {
+  return AtomicAdjust::dec(_ref_count);
+}
+
 /**
  *
  */
diff --git a/dtool/src/prc/streamWrapper.h b/dtool/src/prc/streamWrapper.h
index b5a4d5bf72..a837540129 100644
--- a/dtool/src/prc/streamWrapper.h
+++ b/dtool/src/prc/streamWrapper.h
@@ -16,6 +16,7 @@
 
 #include "dtoolbase.h"
 #include "mutexImpl.h"
+#include "atomicAdjust.h"
 
 /**
  * The base class for both IStreamWrapper and OStreamWrapper, this provides
@@ -30,8 +31,17 @@ PUBLISHED:
   INLINE void acquire();
   INLINE void release();
 
+public:
+  INLINE void ref() const;
+  INLINE bool unref() const;
+
 private:
   MutexImpl _lock;
+
+  // This isn't really designed as a reference counted class, but it is useful
+  // to treat it as one when dealing with substreams created by Multifile.
+  mutable AtomicAdjust::Integer _ref_count = 1;
+
 #ifdef SIMPLE_THREADS
   // In the SIMPLE_THREADS case, we need to use a bool flag, because MutexImpl
   // defines to nothing in this case--but we still need to achieve a form of
diff --git a/panda/src/express/multifile.cxx b/panda/src/express/multifile.cxx
index e625861f87..cdb1c56ec8 100644
--- a/panda/src/express/multifile.cxx
+++ b/panda/src/express/multifile.cxx
@@ -333,7 +333,10 @@ close() {
   if (_owns_stream) {
     // We prefer to delete the IStreamWrapper over the ostream, if possible.
     if (_read != nullptr) {
-      delete _read;
+      // Only delete it if no SubStream is still referencing it.
+      if (!_read->unref()) {
+        delete _read;
+      }
     } else if (_write != nullptr) {
       delete _write;
     }
diff --git a/panda/src/express/subStreamBuf.cxx b/panda/src/express/subStreamBuf.cxx
index 4d773e359f..14a05f1bfb 100644
--- a/panda/src/express/subStreamBuf.cxx
+++ b/panda/src/express/subStreamBuf.cxx
@@ -88,6 +88,13 @@ open(IStreamWrapper *source, OStreamWrapper *dest, streampos start, streampos en
   _append = append;
   _gpos = _start;
   _ppos = _start;
+
+  if (source != nullptr) {
+    source->ref();
+  }
+  if (dest != nullptr) {
+    dest->ref();
+  }
 }
 
 /**
@@ -98,6 +105,12 @@ close() {
   // Make sure the write buffer is flushed.
   sync();
 
+  if (_source != nullptr && !_source->unref()) {
+    delete _source;
+  }
+  if (_dest != nullptr && !_dest->unref()) {
+    delete _dest;
+  }
   _source = nullptr;
   _dest = nullptr;
   _start = 0;
diff --git a/panda/src/express/subStreamBuf.h b/panda/src/express/subStreamBuf.h
index 779f43d94c..1f0c511e51 100644
--- a/panda/src/express/subStreamBuf.h
+++ b/panda/src/express/subStreamBuf.h
@@ -23,6 +23,7 @@
 class EXPCL_PANDA_EXPRESS SubStreamBuf : public std::streambuf {
 public:
   SubStreamBuf();
+  SubStreamBuf(const SubStreamBuf &copy) = delete;
   virtual ~SubStreamBuf();
 
   void open(IStreamWrapper *source, OStreamWrapper *dest, std::streampos start, std::streampos end, bool append);