diff --git a/Library/Homebrew/extend/pathname.rb b/Library/Homebrew/extend/pathname.rb
index 28abd8b0a0..7b6cb276c1 100644
--- a/Library/Homebrew/extend/pathname.rb
+++ b/Library/Homebrew/extend/pathname.rb
@@ -451,6 +451,11 @@ class Pathname
   def dylib?
     false
   end
+
+  sig { returns(T::Array[String]) }
+  def rpaths
+    []
+  end
 end
 
 require "extend/os/pathname"
diff --git a/Library/Homebrew/linkage_checker.rb b/Library/Homebrew/linkage_checker.rb
index 116abd76b1..db753fc9b0 100644
--- a/Library/Homebrew/linkage_checker.rb
+++ b/Library/Homebrew/linkage_checker.rb
@@ -32,6 +32,7 @@ class LinkageChecker
     @unnecessary_deps = []
     @unwanted_system_dylibs = []
     @version_conflict_deps = []
+    @files_missing_rpaths = []
 
     check_dylibs(rebuild_cache: rebuild_cache)
   end
@@ -46,6 +47,7 @@ class LinkageChecker
     display_items "Undeclared dependencies with linkage", @undeclared_deps
     display_items "Dependencies with no linkage", @unnecessary_deps
     display_items "Unwanted system libraries", @unwanted_system_dylibs
+    display_items "Files with missing rpath", @files_missing_rpaths
   end
 
   def display_reverse_output
@@ -69,11 +71,12 @@ class LinkageChecker
     display_items "Unwanted system libraries", @unwanted_system_dylibs, puts_output: puts_output
     display_items "Conflicting libraries", @version_conflict_deps, puts_output: puts_output
     display_items "Undeclared dependencies with linkage", @undeclared_deps, puts_output: puts_output if strict
+    display_items "Files with missing rpath", @files_missing_rpaths, puts_output: puts_output
   end
 
   sig { params(strict: T::Boolean).returns(T::Boolean) }
   def broken_library_linkage?(strict: false)
-    issues = [@broken_deps, @unwanted_system_dylibs, @version_conflict_deps]
+    issues = [@broken_deps, @unwanted_system_dylibs, @version_conflict_deps, @files_missing_rpaths]
     issues << @undeclared_deps if strict
     [issues, unexpected_broken_dylibs, unexpected_present_dylibs].flatten.any?(&:present?)
   end
@@ -152,9 +155,18 @@ class LinkageChecker
     checked_dylibs = Set.new
 
     keg_files_dylibs.each do |file, dylibs|
+      file_has_any_rpath_dylibs = T.let(false, T::Boolean)
       dylibs.each do |dylib|
         @reverse_links[dylib] << file
 
+        # Files that link @rpath-prefixed dylibs must include at
+        # least one rpath in order to resolve it.
+        if !file_has_any_rpath_dylibs && (dylib.start_with? "@rpath/")
+          file_has_any_rpath_dylibs = true
+          pathname = Pathname(file)
+          @files_missing_rpaths << file if pathname.rpaths.empty?
+        end
+
         next if checked_dylibs.include? dylib
 
         checked_dylibs << dylib
diff --git a/Library/Homebrew/os/linux/elf.rb b/Library/Homebrew/os/linux/elf.rb
index f151d23b57..73dd1b016e 100644
--- a/Library/Homebrew/os/linux/elf.rb
+++ b/Library/Homebrew/os/linux/elf.rb
@@ -86,12 +86,20 @@ module ELFShim
     elf_type == :executable
   end
 
+  # The runtime search path, such as:
+  # "/lib:/usr/lib:/usr/local/lib"
   def rpath
     return @rpath if defined? @rpath
 
     @rpath = rpath_using_patchelf_rb
   end
 
+  # An array of runtime search path entries, such as:
+  # ["/lib", "/usr/lib", "/usr/local/lib"]
+  def rpaths
+    rpath.split(":")
+  end
+
   def interpreter
     return @interpreter if defined? @interpreter