it's true that you don't NEED compression or EQ, but it can often help you get a better result. i personally don't really use compression when recording individual instruments, unless it's an instrument that's very dynamic.
EQ'ing to tape is definitely something to try. if you cut out bass frequencies while recording, you can hit the tape harder and the highs will saturate better ... i also found that that said, i don't generally EQ to tape myself, i prefer to bypass the mixer and plug the preamp into the tape machine. but because of tape head bump, i often engage a low cut on the preamp or mic if possible. and often also cut even more bass when mixing.
since you have no compressor, you may have to ride your faders on the vocals while mixing. i don't like the sound of a compressor on regular instruments, but i always use it on vocals. since you mentioned that everything else sounds fine, and the backup vocals are "lost", it still seems like an EQ issue to me.