You certainly get organisms that produce only short stretches of SC where the obligate CO is located, whilst the rest of the chromosome remains unsynapsed (eg Fritillaria and grashopper). Whether there are higher organisms with no SC that maintain the obligate CO, I don't know.

Thank you for information about fritillaries and grashoppers!

I have been considering the possibility of recombination without SC's in higher organisms. As it occurs e.g. in S. pombe. 

I don't think the S. Pombe approach would be reliable enough for higher organisms in general, but there are lots of weird and wonderful creatures that we know nothing about, so something similar could exist in nature. S. pombe has the 'linear elements' but not the central element, so is not so diverged. However, as it lacks the class I pathway of COs, this would be a disadvantage to most eukaryotes.

Male crossing over in Drosophila ananassae is a case, apparently, that does not require SC. Chiasma are observed in diplotene bivalent at counts that correlate with observed recombination frequency. At MI homologs are paired side by side, exchange can be detected as a X-type chiasmata, whereas exchange in inverted regions produce U-type inversion chiasmata.

Goñi B, Matsuda M and YN Tobari 2006. Chiasmata and chromosome breakages are related events of crossing over in males of Drosophila ananassae. Genome 49 (11):1374-1383. Doi: 10.1139/g06-106