Interpreting rel. dS units between HYPHY and PAML

Dears,

I have a dataset for which I have tested the Local/FR and Global/M0
models of codons in HYPHY and PAML for the Codon Frequency models that
they share in their implementation, and found that:

Both programs give exactly (to the fourth/fifth decimal) the same
results, in the Likelihood Value, R/omegas, dN/dS values, tree
branch lengths, etc.

Both programs use, as expected, different units in the branch lengths
of the estimated tree, in a relation of 3 to 1 or to number of changes
per codon or per nucleotide.

Both programs use different units in the dS (and dN for Local/FR)
branch lengths that don't follow a 3 to 1 relationship, and this
relation changes between the different shared Codon Frequency models
or between subsets of the data.

I would like to know how to interpret this difference in units in
absolute terms between the programs.

Here an example: ~2.09 to 1 relation

Global/M0 =

                HYPHY / PAML
                      /
Tkodak.synRate=5.61358 / 2.6837 2.0917316
Pfurio.synRate=2.22195 / 1.0622 2.0918377
Node4.synRate=1.13503 / 0.5426 2.0918356
Pabyss.synRate=1.88494 / 0.9010 2.0920533
Phorik.synRate=1.64998 / 0.7887 2.0920249

Local/FR =

                     HYPHY / PAML
                           /
TKODAK.nonSynRate=0.338324  / 0.1619 2.0897097   ; =0.0480279146887137
TKODAK.synRate   =7.04432   / 3.3721 2.0890009                
PFURIO.nonSynRate=0.141819  / 0.0679 2.0886451   ; =0.0569279865125241
PFURIO.synRate   =2.4912    / 1.1924 2.0892318               
Node4.nonSynRate=0.0775024 / 0.0371 2.0890135    ; =0.898580634389877
Node4.synRate   =0.0862498 / 0.0413 2.0883729                
PABYSS.nonSynRate=0.110662  / 0.0530 2.0879623   ; =0.0519508199030106
PABYSS.synRate   =2.13013   / 1.0196 2.0891820                
PHORIK.nonSynRate=0.10045   / 0.0481 2.0883576  ; =0.0600953628754839
PHORIK.synRate   =1.67151   / 0.8001 2.0891264                

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Due to the constraints of the forum thread, I've emailed more details on the analysis to Sergei emails account.

Thanks in advance,

Albert.