What to even do with pooled 16S and ITS amplicon sequences with no barcodes?

[Michael.Ante]

Hi,
you might use the primer sequences (which might be obtained from the company) to split the data in an ITS and a 16S part which can be analysed separately. Run e.g. cutadapt on the data set with the ITS primer and save the untrimmed reads in a file, which you use as an input for the trimming with the 16S primers.

[roobcf]

Hey there
I've been working exclusively with bioinformatics for a few years and following DNA extraction, I leave all my sequencing needs to be solved by sequencing facilities. Most of my projects require both 16S and ITS sequencing and in most cases, I've received the data in separate runs. Recently, I decided to evaluate a recent company from my country and they sent me the data for both amplicons in the same file. I thought that maybe the samples were multiplexed, but to my surprise, they said they just pool the libraries and run them without indexing per target.
Is there even a correct way to deal with this? I use a DADA2-based workflow, I thought about maybe doing a preliminary classification of each read (using just blast), then separating the targets and following my usual workflow, but this feels like a workaround I should not be taking.