docs: add MLflow AI Gateway example

examples/mlflow_gateway.py

@@ -0,0 +1,68 @@
+#!/usr/bin/env python3
+"""Example of using Claude Agent SDK with MLflow AI Gateway.
+
+MLflow AI Gateway (MLflow >= 3.0) is a database-backed LLM proxy that routes
+requests to multiple providers through a unified API. It provides an
+Anthropic-compatible passthrough at /gateway/anthropic/v1, so the Claude
+Agent SDK can use it as a drop-in replacement for the Anthropic API.
+
+Setup:
+    1. pip install mlflow[genai]
+    2. mlflow server --host 127.0.0.1 --port 5000
+    3. Create a gateway endpoint in the MLflow UI at http://localhost:5000
+       (AI Gateway → Create Endpoint, select Anthropic as provider)
+    4. Set environment variables:
+       export ANTHROPIC_BASE_URL="http://localhost:5000/gateway/anthropic"
+       export ANTHROPIC_API_KEY="unused"  # provider keys are on the server
+
+Usage:
+    python examples/mlflow_gateway.py
+"""
+
+import anyio
+
+from claude_agent_sdk import (
+    AssistantMessage,
+    ClaudeAgentOptions,
+    ResultMessage,
+    TextBlock,
+    query,
+)
+
+
+async def gateway_example():
+    """Use Claude Agent SDK through MLflow AI Gateway."""
+    print("=== MLflow AI Gateway Example ===")
+
+    # Point the SDK to MLflow AI Gateway's Anthropic-compatible passthrough.
+    # The gateway manages provider API keys — ANTHROPIC_API_KEY is not
+    # validated but must be set to a non-empty value.
+    options = ClaudeAgentOptions(
+        system_prompt="You are a helpful assistant. Keep answers concise.",
+        max_turns=1,
+        env={
+            "ANTHROPIC_BASE_URL": "http://localhost:5000/gateway/anthropic",
+            "ANTHROPIC_API_KEY": "unused",
+        },
+    )
+
+    async for message in query(
+        prompt="What is MLflow AI Gateway? Answer in one sentence.",
+        options=options,
+    ):
+        if isinstance(message, AssistantMessage):
+            for block in message.content:
+                if isinstance(block, TextBlock):
+                    print(f"Claude: {block.text}")
+        elif isinstance(message, ResultMessage) and message.total_cost_usd > 0:
+            print(f"\nCost: ${message.total_cost_usd:.4f}")
+
+    print()
+
+
+async def main():
+    await gateway_example()
+
+
+if __name__ == "__main__":
+    anyio.run(main)