Yes, it is possible. My earliest diagnosis was Par ,Schizo. I was happily married (common-law) for 7 years. My son will be 22 this year My recovery has been lengthy but in that time I've had 4 other relationships , two with par.schiz. women.
You speak of health and by that I can assume you mean emotional health and what people would mean when they say"healthy relationship". If you are the caregiver in the relationship, that is , if you are acting in the medical interest of the individual, a romantic relationship may not be advisable, However, if the two of you are in love , then nothing can stop you !! Remember to be patient (pun intended) with your partner . Bear in mind that their world may not connect at all with yours , the real world. And know that recovery is possible.
P.S. re "caregiver" Most couples united this way have a partner who is also involved with medical matters.
Sincerely,
Don Fraser