Husband and Wife both have health insurance from separate employers, but only use Husband's for medical claims.

My wife recently accepted a job at which she will receive a 175% match for all HSA contributions, but she must be enrolled in her new employer's HDHP plan to obtain this benefit. 

For this reason, we would like to enroll her in her new employer's HDHP plan to receive the HSA match, but also keep her on my medical plan as the family deductible is much lower. The math works out such that even with her premium in consideration we will still be net-positive with her 175% HSA match. 

We never intend to submit under her plan for any sort of medical claim, but rather always use mine. We will only use her plan for the HSA match and nothing more. We will use both hers and my HSA accounts to pay the deductible down on my HDHP plan since it will be the only plan we will claim against for medical expenses. 

Being that no medical claims will ever be made against her policy will mine be considered her primary insurance? I do not know if once a claim is submitted for her under my insurance if there is a way for my insurance to find out that she is actually also covered by her own employer.