Basically that's a limitation of ASIO (and ASIO4ALL) when there's no dedicated ASIO driver available from the manufacturer which provides multi-client support.
So try using WASAPI (shared mode) as audio driver in Reaper instead.
You could also have a look at the Steinberg ASIO Multiclient:
https://www.steinberg.net/forums/viewtopic.php?t=48
But in this case the other applications you want to use at the same time (web browser, etc.) need to support ASIO as audio driver, as well.