docs: Document async SDK usage

[marandaneto]

💡 Motivation and Context

Finishes the async SDK stack for #103 with README guidance for installing posthog[async], using AsyncPosthog, closing it with async with / await shutdown(), and evaluating flags without blocking the event loop.

This PR is stacked on #720. Review feedback was addressed by removing a redundant explicit flush() from the async with example, since exiting the context shuts down and flushes automatically.

💚 How did you test it?

• uv run --extra test pytest posthog/test/test_async_request.py posthog/test/test_async_client.py posthog/test/test_async_feature_flags.py --verbose --timeout=30
• uv run python -W error -c "import posthog; from posthog import AsyncPosthog"

📝 Checklist

• I reviewed the submitted code.
• I added tests to verify the changes.
• I updated the docs if needed.
• No breaking change or entry added to the changelog.

If releasing new changes

• Ran sampo add to generate a changeset file

🤖 Agent context

Autonomy: Human-driven (agent-assisted)

Implemented with pi. This PR is docs-only and is intentionally separated from the implementation PRs so the code review can happen in smaller stacked chunks.

[greptile-apps]

_{Reviews (1): Last reviewed commit: "Document async SDK usage" | Re-trigger Greptile}

[github-actions]

posthog-python Compliance Report

Date: 2026-07-02 14:19:04 UTC
Duration: 177037ms

✅ All Tests Passed!

46/46 tests passed

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Capture Tests

✅ 29/29 tests passed

View Details

Test	Status	Duration
Format Validation.Event Has Required Fields	✅	513ms
Format Validation.Event Has Uuid	✅	1506ms
Format Validation.Event Has Lib Properties	✅	1506ms
Format Validation.Distinct Id Is String	✅	1506ms
Format Validation.Token Is Present	✅	1505ms
Format Validation.Custom Properties Preserved	✅	1505ms
Format Validation.Event Has Timestamp	✅	1505ms
Retry Behavior.Retries On 503	✅	9512ms
Retry Behavior.Does Not Retry On 400	✅	3508ms
Retry Behavior.Does Not Retry On 401	✅	3506ms
Retry Behavior.Respects Retry After Header	✅	9513ms
Retry Behavior.Implements Backoff	✅	23525ms
Retry Behavior.Retries On 500	✅	7501ms
Retry Behavior.Retries On 502	✅	7512ms
Retry Behavior.Retries On 504	✅	7509ms
Retry Behavior.Max Retries Respected	✅	23526ms
Deduplication.Generates Unique Uuids	✅	1495ms
Deduplication.Preserves Uuid On Retry	✅	7510ms
Deduplication.Preserves Uuid And Timestamp On Retry	✅	14514ms
Deduplication.Preserves Uuid And Timestamp On Batch Retry	✅	7514ms
Deduplication.No Duplicate Events In Batch	✅	1502ms
Deduplication.Different Events Have Different Uuids	✅	1506ms
Compression.Sends Gzip When Enabled	✅	1506ms
Batch Format.Uses Proper Batch Structure	✅	1506ms
Batch Format.Flush With No Events Sends Nothing	✅	1004ms
Batch Format.Multiple Events Batched Together	✅	1504ms
Error Handling.Does Not Retry On 403	✅	3508ms
Error Handling.Does Not Retry On 413	✅	3506ms
Error Handling.Retries On 408	✅	7512ms

Feature_Flags Tests

✅ 17/17 tests passed

View Details

Test	Status	Duration
Request Payload.Request With Person Properties Device Id	✅	1001ms
Request Payload.Flags Request Uses V2 Query Param	✅	1006ms
Request Payload.Flags Request Hits Flags Path Not Decide	✅	1005ms
Request Payload.Flags Request Omits Authorization Header	✅	1005ms
Request Payload.Token In Flags Body Matches Init	✅	1006ms
Request Payload.Groups Round Trip	✅	1005ms
Request Payload.Groups Default To Empty Object	✅	1005ms
Request Payload.Disable Geoip False Propagates As Geoip Disable False	✅	1006ms
Request Payload.Disable Geoip Omitted Defaults To False	✅	1005ms
Request Payload.Flag Keys To Evaluate Contains Only Requested Key	✅	1006ms
Request Lifecycle.No Flags Request On Init Alone	✅	502ms
Request Lifecycle.No Flags Request On Normal Capture	✅	1506ms
Request Lifecycle.Two Flag Calls Produce Two Remote Requests	✅	1008ms
Request Lifecycle.Mock Response Value Is Returned To Caller	✅	1003ms
Retry Behavior.Retries Flags On 502	✅	1005ms
Retry Behavior.Retries Flags On 504	✅	1005ms
Side Effect Events.Get Feature Flag Captures Feature Flag Called Event	✅	1507ms