There will happen nothing, because cliens have a redundant connection to each RR, this way, if a internet connection fails, or a RR fails, or the client to a RR path fail, there will be a exit/incoming path through the other RR.
Think that if you peer your RR, there is only one peer more to get a full mesh ibgp, from both clients. Of course if you have 10 ibgp clients this is not the case.
If clients cant have a ibgp connection with each RR, a redundant connection, or you need the incoming traffic to get into your AS through a specific internet connection for some prefix, and others prefix through the other connection, you will need to peer both RR. This way a client will have always a incoming path from both internet connections.
You have to be carefull with which routers propagate the prefixes. if you learn and propagate from your ibgp, will need redundant ibgp connections, both client to RR, or RR to RR. This way will assure prefix are propagated outside.
If you use your ebgp routers to propagate internal prefixes, have to be carefull to dont lost connection with its internal routers. Because you will propagate something you cant reach. In this case, if you lost your RR connection and each client dont peer with each RR, then will lost the path to internal peers from the ebgp router. In this case you will need to peer RR and each client with both RR.
Or use a conditional route advertisement, to avoid propagate prefix you cant reach from that router.
There is many cases, depends on how you use your internet connections, and from where you propagate the prefix. There should be always a redundant path, and have to be sure to dont propagate unreachable paths.