Hi @Daniela and welcome to the forum 
From the little I’ve just read on the internet, your partner’s dizziness could be a symptom of the brain stem compression.
But it could also be from any number of things including the stroke in general. I too had dizziness/giddiness and fatigue in the early months post stroke (which were TIA’s). The dizziness righted itself in the first year, the giddiness was nearer 2+yrs, I’m now nearly 4yrs post stroke.
I suggest he discusses these symptoms with his gp being linked to the brain stem compression and find out whether it needs further investigation. Just don’t dismiss it as a stroke symptom that will hopefully correct itself over time, not until all other avenues have been investigated and ruled out 