The affect on differential backups when Protecting SQL DB with multiple Backups products at the same time

I am wondering what happens to the SQL full backup and differential backup relationship in one backup product if another backup product is also doing full backups on a daily basis.

We are using one backup solution to protect the sql server locally on a daily basis and this solution performs a full SQL backup.  We also have a second backup solution that backups up the SQL DB offsite to the cloud and it performs a full backup once a week and a differential backup everyday.  What happens to the Full backup Differential backup relationship of the cloud solution if another solution is creating full backups.  Which system keeps track of the backup LSN numbers, is it SQL or the backup solutions?

For example if I try to do a restore of the cloud backup solution will the differential backup restore work or will it be linked to the on premise full backup as apposed to the cloud full backup?