To build on Louis's helpful answer...
from selenium import webdriver
from selenium.webdriver.common.desired_capabilities import DesiredCapabilities
...
caps = DesiredCapabilities.PHANTOMJS
caps["phantomjs.page.settings.userAgent"] = "whatever you want"
driver = webdriver.PhantomJS(desired_capabilities=caps)
The only minor issue is that, unlike for Firefox and Chrome, this does not return your custom setting:
driver.execute_script("return navigator.userAgent")
So, if anyone figures out how to do that in PhantomJS, please edit my answer or add a comment below! Cheers.