You could say that the payments are aligned to calendar months, but normally you would need to say something like
payments are made monthly on the same day of each month, which may be the day of the month on which the tenancy started or the first day of the month.
Completion to calendar month might (in English) indicate the payment for the part-month which needs to be paid for separately in order that future payments are aligned to calendar months. Thus for a tenant starting on 20th, you would have a completion payment to "complete" the partial month.
However, as completion usually means something else in contract law (the final signing and seal) it's not a good phrase to use here.
Tenancies in England always run in exact months from the date they start, so a payment made on the first day of the month is paying for the previous part-month in arrears and the next part-month in advance. Consequently in England we don't have your concept of completion and this makes a direct translation difficult.
When I took out my mortgage, the mortgage company wanted payments made in advance on the first day of the month, so the first payment included the previous two weeks as well1. This was actually called arrears:
Money that is owed and should have been paid earlier
...even though it was never actually treated as late. Your "completion" payment to align future payments to calendar months, if it's made late, is technically a payment in arrears.
If the tenant simply pays the first couple of weeks in advance, and then makes monthly payments after that, then a payment to align to calendar months describes it fairly succinctly.
1 And the final payment was less as it only went to the anniversary date, not the full calendar month.