diff --git a/packages/eslint-plugin/src/rules/no-floating-promises.ts b/packages/eslint-plugin/src/rules/no-floating-promises.ts
index b3ac65296992..b9fb9b1cc4dd 100644
--- a/packages/eslint-plugin/src/rules/no-floating-promises.ts
+++ b/packages/eslint-plugin/src/rules/no-floating-promises.ts
@@ -290,6 +290,8 @@ export default createRule<Options, MessageId>({
 
         // All other cases are unhandled.
         return { isUnhandled: true };
+      } else if (node.type === AST_NODE_TYPES.TaggedTemplateExpression) {
+        return { isUnhandled: true };
       } else if (node.type === AST_NODE_TYPES.ConditionalExpression) {
         // We must be getting the promise-like value from one of the branches of the
         // ternary. Check them directly.
diff --git a/packages/eslint-plugin/tests/rules/no-floating-promises.test.ts b/packages/eslint-plugin/tests/rules/no-floating-promises.test.ts
index 434c9735a46d..bd02ed6d5a87 100644
--- a/packages/eslint-plugin/tests/rules/no-floating-promises.test.ts
+++ b/packages/eslint-plugin/tests/rules/no-floating-promises.test.ts
@@ -504,6 +504,18 @@ void promiseArray;
 ['I', 'am', 'just', 'an', 'array'];
       `,
     },
+    {
+      code: `
+declare const myTag: (strings: TemplateStringsArray) => Promise<void>;
+myTag\`abc\`.catch(() => {});
+      `,
+    },
+    {
+      code: `
+declare const myTag: (strings: TemplateStringsArray) => string;
+myTag\`abc\`;
+      `,
+    },
   ],
 
   invalid: [
@@ -593,6 +605,43 @@ doSomething();
         },
       ],
     },
+    {
+      code: `
+declare const myTag: (strings: TemplateStringsArray) => Promise<void>;
+myTag\`abc\`;
+      `,
+      errors: [
+        {
+          line: 3,
+          messageId: 'floatingVoid',
+        },
+      ],
+    },
+    {
+      code: `
+declare const myTag: (strings: TemplateStringsArray) => Promise<void>;
+myTag\`abc\`.then(() => {});
+      `,
+      errors: [
+        {
+          line: 3,
+          messageId: 'floatingVoid',
+        },
+      ],
+    },
+    {
+      code: `
+declare const myTag: (strings: TemplateStringsArray) => Promise<void>;
+myTag\`abc\`.finally(() => {});
+      `,
+      errors: [
+        {
+          line: 3,
+          messageId: 'floatingVoid',
+        },
+      ],
+    },
+
     {
       options: [{ ignoreVoid: true }],
       code: `