diff --git a/coverage/phystokens.py b/coverage/phystokens.py
index 9fc36ecda..0c344b0af 100644
--- a/coverage/phystokens.py
+++ b/coverage/phystokens.py
@@ -57,10 +57,11 @@ def _phys_tokens(toks: TokenInfos) -> TokenInfos:
                 if last_ttext.endswith("\\"):
                     inject_backslash = False
                 elif ttype == token.STRING:
-                    if last_line.endswith(last_ttext + "\\\n"):
+                    if (last_line.endswith("\\\n") and
+                        last_line.rstrip(" \\\n").endswith(last_ttext)):
                         # Deal with special cases like such code::
                         #
-                        #   a = ["aaa",\
+                        #   a = ["aaa",\ # there may be zero or more blanks between "," and "\".
                         #        "bbb \
                         #        ccc"]
                         #
@@ -69,6 +70,9 @@ def _phys_tokens(toks: TokenInfos) -> TokenInfos:
                         # It's a multi-line string and the first line ends with
                         # a backslash, so we don't need to inject another.
                         inject_backslash = False
+                elif sys.version_info >= (3, 12) and ttype == token.FSTRING_MIDDLE:
+                    if ttext.split("\n", 1)[0][-1] == "\\":
+                        inject_backslash = False
                 if inject_backslash:
                     # Figure out what column the backslash is in.
                     ccol = len(last_line.split("\n")[-2]) - 1
diff --git a/tests/test_html.py b/tests/test_html.py
index d5cc0f301..5b5e3b86c 100644
--- a/tests/test_html.py
+++ b/tests/test_html.py
@@ -1203,6 +1203,36 @@ def test_bug_1828(self) -> None:
             '3     ccc"]',
             ]
 
+    @pytest.mark.parametrize(
+        "leader", ["", "f", "r", "fr", "rf"],
+        ids=["string", "f-string", "raw_string", "f-raw_string", "raw_f-string"]
+    )
+    def test_bug_1836(self, leader: str) -> None:
+        # https://github.com/nedbat/coveragepy/issues/1836
+        self.make_file("py312_fstrings.py", f"""\
+            prog_name = 'bug.py'
+            err_msg = {leader}'''\\
+            {{prog_name}}: ERROR: This is the first line of the error.
+            {{prog_name}}: ERROR: This is the second line of the error.
+            \\
+            {{prog_name}}: ERROR: This is the third line of the error.
+            '''
+            """)
+
+        cov = coverage.Coverage()
+        py312_fstrings = self.start_import_stop(cov, "py312_fstrings")
+        cov.html_report(py312_fstrings)
+
+        assert self.get_html_report_text_lines("py312_fstrings.py") == [
+            "1" + "prog_name = 'bug.py'",
+            "2" + f"err_msg = {leader}'''\\",
+            "3" + "{prog_name}: ERROR: This is the first line of the error.",
+            "4" + "{prog_name}: ERROR: This is the second line of the error.",
+            "5" + "\\",
+            "6" + "{prog_name}: ERROR: This is the third line of the error.",
+            "7" + "'''",
+            ]
+
     def test_unicode(self) -> None:
         surrogate = "\U000e0100"