The Anthropic client needs to be closed:

import asyncio
from time import perf_counter

import numpy as np
from anthropic import Anthropic, AsyncAnthropic


async def coroutine1():
    prompt = "Who discovered gravity. Answer in 10 words."

    client = AsyncAnthropic()
    message = await client.messages.create(
        model="claude-3-5-sonnet-20240620",
        max_tokens=1024,
        messages=[{"role": "user", "content": prompt}],
    )
    await client.close()

    return message.content[0].text


async def coroutine2():
    prompt = "Who discovered radioactivity. Answer in 10 words."

    client = AsyncAnthropic()
    message = await client.messages.create(
        model="claude-3-5-sonnet-20240620",
        max_tokens=1024,
        messages=[{"role": "user", "content": prompt}],
    )
    await client.close()

    return message.content[0].text


async def main():
    async with asyncio.TaskGroup() as tg:
        t0 = tg.create_task(coroutine1())
        t1 = tg.create_task(coroutine2())
    print(t0.result())
    print(t1.result())


if __name__ == "__main__":
    for i in range(10):
        tic = perf_counter()
        asyncio.run(main())
        toc = perf_counter()
        print(f"Elapsed time = {toc - tic:.3f}")