It is customary to use headphones for studio tracking for monitoring precisely to be able to record isolated tracks which gives you more ability to manipulate things (both sonically and arrangement alterations).
Some artists however do prefer to record 'live' in the control room (Bono is a name you may have heard of). If you don't plan on replacing and/or changing the part (ie. going for a live take just like a live show), a little bleed from the other instruments isn't going to be much of a problem. After all, live performances can be mixed to full studio quality sonics if the show was run with classy equipment (ie. nice mics - put the 58's away).
If this is all new to you and you simply tried it this way because you didn't consider headphones... just try some headphones. Most people find this method of studio tracking easiest.
My tip of the day is to suggest your singer use their in-ear monitor buds if they have a professional set (not cheap ear-buds). Magically cuts down on the headphone bleed!
The phase trick is a tried and true thing. The 3rd point of that equilateral triangle is of course the engineers chain in the control room. Engineer can grab his iPad and run the computer with remote desktop if needed.