Hi,
So you have 2 routers having an iBGP peering together, and each is having an eBGP peering to the same customer, right?
What are you receiving from your external neighbor on each routers?
What your routers are transmitting/receiving in terms of routes?
Is the customer using a same ASN?
Do you set any Local Pref?
Are you changing your nexthop for your iBGP peering?