From the Guidelines

Calcium supplements can decrease phosphate levels by forming insoluble complexes in the intestinal tract, which reduces phosphate absorption.

Phosphate supplements should not be given together with calcium supplements or foods with high calcium content, such as milk, as precipitation in the intestinal tract reduces absorption 1.
Nutritional calcium intake should be kept within the normal range for age, and calcium supplements are not recommended given that bone mass and mineral content are usually not decreased and because of the potential risk of hypercalciuria 1.
The use of calcium supplements can lead to hypercalciuria, which may promote nephrocalcinosis 1.

Phosphate supplements should be taken as frequently as possible, for example, 4–6 times daily in young patients with high alkaline phosphatase (ALP) levels 1.
Active vitamin D (calcitriol or alfacalcidol) is given in addition to oral phosphate supplements to counter calcitriol deficiency, prevent secondary hyperparathyroidism, and increase phosphate absorption from the gut 1.
Native vitamin D (cholecalciferol or ergocalciferol) should be supplemented in case of vitamin D deficiency 1.

From the Research

Calcium supplements can affect phosphate levels in the body, with studies showing that the form and level of phosphorus intake, as well as the level of calcium intake, can impact calcium metabolism and phosphate balance 2, 3, 4, 5, 6.
The absorption of calcium and phosphate from supplements such as tricalcium phosphate (TCP) and calcium carbonate (CC) has been compared, with results showing that TCP is absorbed adequately and does not stimulate parathyroid activity despite a rise in serum phosphorus levels 4.
The effect of calcium supplements on phosphate balance has been studied in adolescent girls, with results showing that a calcium dietary supplement does not affect phosphorus balance when phosphorus intake is at a typical US intake level of 1400 mg/d 6.
The interaction between vitamin D and phosphate has also been studied, with results showing that vitamin D plays an essential role in maintaining optimal levels of calcium and inorganic phosphate for bone mineralization, and that serum phosphate levels can influence vitamin D activation and parathyroid hormone release 2.
The importance of a balanced calcium and phosphorus intake has been highlighted, with excessive urinary phosphate loss and hyperphosphatemia being risk factors for cardiovascular calcification, chronic kidney diseases, and premature aging 2, 3.

Calcium supplements can increase faecal calcium and phosphorus, and urinary calcium, while decreasing urinary phosphorus and increasing postprandial calcium concentrations 3, 4.
Phosphorus supplements can cause a reduction in fractional renal tubular reabsorption of calcium and an increase in urinary cyclic AMP, indicating an increase in parathyroid hormone secretion 5.
A combined supplement of calcium and orthophosphate can cause decreases in the excretion of both cyclic AMP and hydroxyproline, suggesting a decrease in parathyroid hormone-mediated bone resorption 5.

A study comparing the effects of supplementation with calcium phosphate or phosphate alone in healthy humans found that calcium phosphate supplementation resulted in positively modulated blood lipids and gut-related parameters, while solely phosphate supplementation increased urinary phosphorus and decreased urinary calcium 3.
A study on the absorption of tricalcium phosphate found that it is absorbed adequately and does not stimulate parathyroid activity despite a rise in serum phosphorus levels 4.
A study on phosphorus balance in adolescent girls found that a calcium dietary supplement does not affect phosphorus balance when phosphorus intake is at a typical US intake level of 1400 mg/d 6.