Tenant wants to add a roommate

[Christian Mani]

Hello all, new member's first post here. I'll appreciate any input you might have for us on this tenant question. I apologize if this has been covered ad nauseam here. 

My wife and I currently are renting out our first direct involvement real estate investment. It's a 3/2 SFH in Walnut Creek, CA (30 mins east of SF). We bought and fixed it up in the latter half of 2016 and it's been rented ever since to our first tenant(s). Our two tenants were both professionals (nurse/nurse manager) making a combined income of around $350k. They were also boyfriend/girlfriend. Both names were on the original 1 year lease which has since lapsed and the tenancy is now m2m. They have been extremely low impact tenants and have been on time (actually early) with the rent since day one. The boyfriend just informed me via text that the girlfriend moved out several months ago and that he has been paying the rent ($3600/mo) by himself and that he would like to add a roommate to help finance the cost of the rent. He said that once he found someone suitable he would submit his/her information to me for a background check.

This all sounds reasonable to me, but I'm not sure how to proceed.  Do I care if he has a roommate as long the rent continues to get paid...in essence allowing him to sublet? Or do I have my current tenant plus the new tenant each sign a new one year lease?   What is the most prudent course given my situation?

Am I up a creek in Walnut Creek?

Thanks in advance for any input on this.

Christian Mani